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## How to Solve Probability Problems? (+FREE Worksheet!)

Do you want to know how to solve Probability Problems? Here you learn how to solve probability word problems.

## Related Topics

- How to Interpret Histogram
- How to Interpret Pie Graphs
- How to Solve Permutations and Combinations
- How to Find Mean, Median, Mode, and Range of the Given Data

## Step by step guide to solve Probability Problems

- Probability is the likelihood of something happening in the future. It is expressed as a number between zero (can never happen) to \(1\) (will always happen).
- Probability can be expressed as a fraction, a decimal, or a percent.
- To solve a probability problem identify the event, find the number of outcomes of the event, then use probability law: \(\frac{number\ of \ favorable \ outcome}{total \ number \ of \ possible \ outcomes}\)

The Absolute Best Books to Ace Pre-Algebra to Algebra II

## The Ultimate Algebra Bundle From Pre-Algebra to Algebra II

Probability problems – example 1:.

If there are \(8\) red balls and \(12\) blue balls in a basket, what is the probability that John will pick out a red ball from the basket?

There are \(8\) red balls and \(20\) a total number of balls. Therefore, the probability that John will pick out a red ball from the basket is \(8\) out of \(20\) or \(\frac{8}{8+12}=\frac{8}{20}=\frac{2}{5}\).

## Probability Problems – Example 2:

A bag contains \(18\) balls: two green, five black, eight blue, a brown, a red, and one white. If \(17\) balls are removed from the bag at random, what is the probability that a brown ball has been removed?

If \(17\) balls are removed from the bag at random, there will be one ball in the bag. The probability of choosing a brown ball is \(1\) out of \(18\). Therefore, the probability of not choosing a brown ball is \(17\) out of \(18\) and the probability of having not a brown ball after removing \(17\) balls is the same.

## Exercises for Solving Probability Problems

The Best Book to Help You Ace Pre-Algebra

## Pre-Algebra for Beginners The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

- A number is chosen at random from \(1\) to \(10\). Find the probability of selecting a \(4\) or smaller.
- A number is chosen at random from \(1\) to \(50\). Find the probability of selecting multiples of \(10\).
- A number is chosen at random from \(1\) to \(10\). Find the probability of selecting of \(4\) and factors of \(6\).
- A number is chosen at random from \(1\) to \(10\). Find the probability of selecting a multiple of \(3\).
- A number is chosen at random from \(1\) to \(50\). Find the probability of selecting prime numbers.
- A number is chosen at random from \(1\) to \(25\). Find the probability of not selecting a composite number.

## Download Probability Problems Worksheet

- \(\color{blue}{\frac{2}{5}}\)
- \(\color{blue}{\frac{1}{10}}\)
- \(\color{blue}{\frac{1}{2}}\)
- \(\color{blue}{\frac{3}{10}}\)
- \(\color{blue}{\frac{9}{25}}\)

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Pre-algebra in 10 days the most effective pre-algebra crash course, college algebra practice workbook the most comprehensive review of college algebra, high school algebra i a comprehensive review and step-by-step guide to mastering high school algebra 1, 10 full length clep college algebra practice tests the practice you need to ace the clep college algebra test.

by: Effortless Math Team about 4 years ago (category: Articles , Free Math Worksheets )

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Teach yourself statistics

## How to Solve Probability Problems

You can solve many simple probability problems just by knowing two simple rules:

- The probability of any sample point can range from 0 to 1.
- The sum of probabilities of all sample points in a sample space is equal to 1.

The following sample problems show how to apply these rules to find (1) the probability of a sample point and (2) the probability of an event.

## Probability of a Sample Point

The probability of a sample point is a measure of the likelihood that the sample point will occur.

Example 1 Suppose we conduct a simple statistical experiment . We flip a coin one time. The coin flip can have one of two equally-likely outcomes - heads or tails. Together, these outcomes represent the sample space of our experiment. Individually, each outcome represents a sample point in the sample space. What is the probability of each sample point?

Solution: The sum of probabilities of all the sample points must equal 1. And the probability of getting a head is equal to the probability of getting a tail. Therefore, the probability of each sample point (heads or tails) must be equal to 1/2.

Example 2 Let's repeat the experiment of Example 1, with a die instead of a coin. If we toss a fair die, what is the probability of each sample point?

Solution: For this experiment, the sample space consists of six sample points: {1, 2, 3, 4, 5, 6}. Each sample point has equal probability. And the sum of probabilities of all the sample points must equal 1. Therefore, the probability of each sample point must be equal to 1/6.

## Probability of an Event

The probability of an event is a measure of the likelihood that the event will occur. By convention, statisticians have agreed on the following rules.

- The probability of any event can range from 0 to 1.
- The probability of event A is the sum of the probabilities of all the sample points in event A.
- The probability of event A is denoted by P(A).

Thus, if event A were very unlikely to occur, then P(A) would be close to 0. And if event A were very likely to occur, then P(A) would be close to 1.

Example 1 Suppose we draw a card from a deck of playing cards. What is the probability that we draw a spade?

Solution: The sample space of this experiment consists of 52 cards, and the probability of each sample point is 1/52. Since there are 13 spades in the deck, the probability of drawing a spade is

P(Spade) = (13)(1/52) = 1/4

Example 2 Suppose a coin is flipped 3 times. What is the probability of getting two tails and one head?

Solution: For this experiment, the sample space consists of 8 sample points.

S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

Each sample point is equally likely to occur, so the probability of getting any particular sample point is 1/8. The event "getting two tails and one head" consists of the following subset of the sample space.

A = {TTH, THT, HTT}

The probability of Event A is the sum of the probabilities of the sample points in A. Therefore,

P(A) = 1/8 + 1/8 + 1/8 = 3/8

Helping math teachers bring statistics to life

- Oct 30, 2021

## 5 Probability Strategies to Try Before Using a Formula

Updated: Aug 29, 2023

In our end-of-the-year survey for our students, there is one message that has been consistently clear for many years: Probability is the worst !

But how can this be? With only two formulas, a student can solve most of the probability questions that can come their way.

## Understanding the Problem

The problem is that students are far too quick to reach for these formulas. We understand why, as this approach may have served them well in other math and science courses. But probability is different.

For understanding probability, memorizing and implementing formulas just doesn’t seem to work for students.

Here are all the ways that using formulas for probability can go wrong for students:

students don’t remember the correct formula

students choose the wrong formula, or the wrong version of the right formula

students make errors in implementing the formulas, often substituting incorrect values

students get a final answer and don’t know whether or not it is reasonable

All of these student mistakes are easy to explain. Students are missing the fundamental understanding about probability ideas and concepts. They don’t do any thinking and reasoning about the context and information given in the problem before jumping right to the formula. Using formulas as a strategy to solve probability questions obscures the conceptual understanding that is needed for success.

## The Solution

Suggest to students that formulas are a last resort. We need to start by trying some strategies that build conceptual understanding of probability concepts. Here are our top 5:

## 1. Simulation

We can use a simulation to estimate a probability by doing many trials of simulating a random phenomena. We know that as the number of trials increases, the proportion of times that our chosen event occurs will approach the true probability (Law of Large Numbers!).

Example: There are 10 balls numbered 1-10 in a bucket and we randomly choose 3 of them. What is the probability of getting a sum of 15 or higher?

When to use it: Use a simulation for a probability that might be really hard to calculate with a formula because of a complex scenario or difficulty in counting outcomes in the event or the sample space.

Stats Medic Lesson: Are Soda Contests True?

## 2. Sample Space

List out all of the events in the sample space. Circle the ones that are in the event for which you want to calculate a probability.

Example: Toss a fair coin 3 times. What is the probability of getting at least 2 tails?

When to use it: Use the sample space when the number of possible outcomes is reasonably small and easy to write out (or think about). Be sure that each outcome is equally likely if you are going to use this approach.

Stats Medic Lesson: The Last Banana

## 3. Two-Way Tables

Sometimes we are given counts or percents for two categorical variables for a sample or a population. These counts or percents can be summarized nicely in a two-way table.

Example: 80% of students at East Kentwood High School have an Instagram account, 60% have a Twitter account, and 45% have both Instagram and Twitter. Given that a student has a Twitter account, what is the probability that they have an Instagram account?

When to use it: Use a two-way table when you have probabilities for two events that are not mutually exclusive. Two-way tables can even be used to find conditional probabilities.

Stats Medic Lesson: Can You Taco Tongue and Evil Eyebrow?

## 4. Venn Diagrams

Sometimes we are given counts or percents for two categorical variables for a sample or a population. While these counts or percents can be summarized nicely in a two-way table, many students prefer the more visual approach of the Venn Diagram.

When to use it: Use a Venn Diagram when you have probabilities for two events that are not mutually exclusive. Venn Diagrams can even be used to find conditional probabilities.

Notice that two-way tables and Venn Diagrams are BFFs .

## 5. Tree Diagrams

A tree diagram is an excellent way to visualize a scenario with several different joint probabilities. When a joint probability can be thought of as one event followed by another event, a tree diagram can help students to keep track of all the possibilities and their respective probabilities.

Example: For AP Statistics class this year, 21% of students are Juniors and 79% are Seniors. Of the Juniors, 55% of them are taking another AP course, while 82% of the Seniors are taking another AP course. Given that a randomly selected AP Stats student is taking another AP course, what is the probability that they are a Senior?

When to use it: Tree diagrams are a great strategy to use when there are multiple events happening (or not happening) in a problem and you are given several conditional probabilities.

Stats Medic Lesson: Can You Get a Pair of Aces or a Pair of Kings?

## A Final Thought

Consider giving students a preemptive pep talk before starting the probability unit. Explain to students why relying on a formula is dangerous and encourage them to embrace the many strategies that will be presented in the upcoming lessons. You could even give them this handout, where they can add their strategies as they learn them (the last blank in the handout is for formulas – a last resort!)

Probability Strategies - WORD

Probability Strategies - pdf

## Recent Posts

After the AP Exam: The Data Science Challenge

A Detailed Analysis of the Questions Used on the AP Statistics Exam

Would This Get Credit? 2023 AP Statistics Exam #4

## Probability Word Problems

In these lessons, we will learn how to solve a variety of probability problems.

Related Pages Probability Tree Diagrams Probability Without Replacement Theoretical vs. Experimental Probability More Lessons On Probability

Here we shall be looking into solving probability word problems involving:

- Probability and Sample Space
- Probability and Frequency Table
- Probability and Area
- Probability of Simple Events
- Probability and Permutations
- Probability and Combinations
- Probability of Independent Events

We will now look at some examples of probability problems.

Example: At a car park there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are lorries. If every vehicle is equally likely to leave, find the probability of: a) a van leaving first. b) a lorry leaving first. c) a car leaving second if either a lorry or van had left first.

Solution: a) Let S be the sample space and A be the event of a van leaving first. n(S) = 100 n(A) = 30

c) If either a lorry or van had left first, then there would be 99 vehicles remaining, 60 of which are cars. Let T be the sample space and C be the event of a car leaving. n(T) = 99 n(C) = 60

Example: A survey was taken on 30 classes at a school to find the total number of left-handed students in each class. The table below shows the results:

A class was selected at random. a) Find the probability that the class has 2 left-handed students. b) What is the probability that the class has at least 3 left-handed students? c) Given that the total number of students in the 30 classes is 960, find the probability that a student randomly chosen from these 30 classes is left-handed.

a) Let S be the sample space and A be the event of a class having 2 left-handed students. n(S) = 30 n(A) = 5

b) Let B be the event of a class having at least 3 left-handed students. n(B) = 12 + 8 + 2 = 22

c) First find the total number of left-handed students:

Total no. of left-handed students = 2 + 10 + 36 + 32 + 10 = 90

Here, the sample space is the total number of students in the 30 classes, which was given as 960.

Let T be the sample space and C be the event that a student is left-handed. n(T) = 960 n(C) = 90

## Probability And Area

Example: ABCD is a square. M is the midpoint of BC and N is the midpoint of CD. A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN.

Area of square = 2x × 2x = 4x 2

This video shows some examples of probability based on area.

## Probability Of Simple Events

The following video shows some examples of probability problems. A few examples of calculating the probability of simple events.

- What is the probability of the next person you meeting having a phone number that ends in 5?
- What is the probability of getting all heads if you flip 3 coins?
- What is the probability that the person you meet next has a birthday in February? (Non-leap year)

This video introduces probability and gives many examples to determine the probability of basic events.

A bag contains 8 marbles numbered 1 to 8 a. What is the probability of selecting a 2 from the bag? b. What is the probability of selecting an odd number? c. What is the probability of selecting a number greater than 6?

Using a standard deck of cards, determine each probability. a. P(face card) b. P(5) c. P(non face card)

## Using Permutations To Solve Probability Problems

This video shows how to evaluate factorials, how to use permutations to solve probability problems, and how to determine the number of permutations with indistinguishable items.

A permutation is an arrangement or ordering. For a permutation, the order matters.

- If a class has 28 students, how many different arrangements can 5 students give a presentation to the class?
- How many ways can the letters of the word PHEONIX be arranged?
- How many ways can you order 3 blue marbles, 4 red marbles and 5 green marbles? Marbles of the same color look identical.

## Using Combinations To Solve Probability Problems

This video shows how to evaluate combinations and how to use combinations to solve probability problems.

A combination is a grouping or subset of items. For a combination, the order does not matter.

- The soccer team has 20 players. There are always 11 players on the field. How many different groups of players can be in the field at the same time?
- A student needs 8 more classes to complete her degree. If she has met the prerequisites for all the courses, how many ways can she take 4 class next semester?
- There are 4 men and 5 women in a small office. The customer wants a site visit from a group of 2 men and 2 women. How many different groups van be formed from the office?

## How To Find The Probability Of Different Events?

This video explains how to determine the probability of different events. This can be found that can be found using combinations and basic probability.

- The probability of drawing 2 cards that are both face cards.
- The probability of drawing 2 cards that are both aces.
- The probability of drawing 4 cards all from the same suite.

A group of 10 students made up of 6 females and 4 males form a committee of 4. What is the probability the committee is all male? What is the probability that the committee is all female? What is the probability the committee is made up of 2 females and 2 males?

## How To Find The Probability Of Multiple Independent Events?

This video explains the counting principle and how to determine the number of ways multiple independent events can occur.

- How many ways can students answer a 3-question true of false quiz?
- How many passwords using 6 digits where the first digit must be letters and the last four digits must be numbers?
- A restaurant offers a dinner special in which you get to pick 1 item from 4 different categories. How many different meals are possible?
- A door lock on a classroom requires entry of 4 digits. All digits must be numbers, but the digits can not be repeated. How many unique codes are possible?

## How To Find The Probability Of A Union Of Two Events?

This video shows how to determine the probability of a union of two events.

- If you roll 2 dice at the same time, what is the probability the sum is 6 or a pair of odd numbers?
- What is the probability of selecting 1 card that is red or a face card?

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## How to Understand Probability

Last Updated: January 2, 2023 Fact Checked

This article was co-authored by Mario Banuelos, PhD . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 9 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 74,688 times.

Knowing how to calculate the probability of an event or events happening can be a valuable skill when making decisions, whether playing a game or in real life. How you calculate probability changes, however, depending on the type of event you are looking to occur. For example, you would not calculate your chances of winning the lottery the same way you would calculate your chances of drawing a full house in a game of poker. Once you determine whether the events are independent, conditional, or mutually exclusive, calculating their probability is very simple.

## Understanding What Probability Means

- Before you can understand more complex probability theory, you must understand how to figure out the probability of a single, random event happening, and understand what that probability means.

## Understanding the Probability of Multiple Independent Events

- For example, if you are using two dice, you might want to know what the probability is that you will roll a double 3. The chance that you will throw a 3 with one die does not affect the chance that you will throw a 3 with the second die, thus the events are independent.

- For a refresher on how to multiply fractions, read Multiply Fractions .

## Understanding the Probability of Conditional Events

- For example, if you are drawing from a standard deck of cards, you might want to know what the probability is of drawing a heart on the first and second draws. Drawing a heart the first time affects the probability of it happening again, because once you draw one heart, there are fewer hearts in the deck, and fewer cards in the deck.

## Understanding the Probability of Mutually Exclusive Events

- Mutually exclusive events will be marked by the conjunction or . (Events that are not mutually exclusive will use the conjunction and .) [12] X Research source
- For example, if you are rolling one die, you might want to know the probability of rolling a 3 or a 4. You cannot roll a 3 and a 4 with one die, so the events are mutually exclusive.

- For a refresher on how to add fractions, read Add Fractions .

## Expert Q&A

## You Might Also Like

- ↑ https://www.mathsisfun.com/data/probability.html
- ↑ http://onlinestatbook.com/2/probability/basic.html
- ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview. 11 December 2020.
- ↑ https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/12.7/related/lesson/calculating-probabilities-of-combined-events-alg-ii/
- ↑ https://www.mathsisfun.com/data/probability-events-independent.html
- ↑ https://www.mathsisfun.com/data/probability-events-conditional.html
- ↑ https://www.mathsisfun.com/data/probability-events-mutually-exclusive.html
- ↑ https://openstax.org/books/statistics/pages/3-2-independent-and-mutually-exclusive-events
- ↑ https://libraryguides.centennialcollege.ca/c.php?g=717168&p=5127684

## About This Article

To understand probability, learn that it refers to the likelihood of an unpredictable event occurring. If you want to calculate the probability of a single event, you'll want to divide the number of favorable outcomes by the number of potential outcomes. For example, if you have 5 blue marbles and 10 red marbles in a box and want to know the probability of you pulling out a blue marble, divide 5 by 15. Since you can simplify 5 divided by 15 to 1 divided by 3, you know that there is a 1 in 3 chance of you pulling out a blue marble. To find out how to calculate the probability of multiple events taking place, keep reading! Did this summary help you? Yes No

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## Unit 7: Probability

About this unit, basic theoretical probability.

- Intro to theoretical probability (Opens a modal)
- Probability: the basics (Opens a modal)
- Simple probability: yellow marble (Opens a modal)
- Simple probability: non-blue marble (Opens a modal)
- Intuitive sense of probabilities (Opens a modal)
- The Monty Hall problem (Opens a modal)
- Simple probability Get 5 of 7 questions to level up!
- Comparing probabilities Get 5 of 7 questions to level up!

## Probability using sample spaces

- Probability with counting outcomes (Opens a modal)
- Example: All the ways you can flip a coin (Opens a modal)
- Die rolling probability (Opens a modal)
- Subsets of sample spaces (Opens a modal)
- Subsets of sample spaces Get 3 of 4 questions to level up!

## Basic set operations

- Intersection and union of sets (Opens a modal)
- Relative complement or difference between sets (Opens a modal)
- Universal set and absolute complement (Opens a modal)
- Subset, strict subset, and superset (Opens a modal)
- Bringing the set operations together (Opens a modal)
- Basic set notation Get 5 of 7 questions to level up!

## Experimental probability

- Experimental probability (Opens a modal)
- Theoretical and experimental probabilities (Opens a modal)
- Making predictions with probability (Opens a modal)
- Simulation and randomness: Random digit tables (Opens a modal)
- Experimental probability Get 5 of 7 questions to level up!
- Making predictions with probability Get 5 of 7 questions to level up!

## Randomness, probability, and simulation

- Experimental versus theoretical probability simulation (Opens a modal)
- Theoretical and experimental probability: Coin flips and die rolls (Opens a modal)
- Random number list to run experiment (Opens a modal)
- Random numbers for experimental probability (Opens a modal)
- Statistical significance of experiment (Opens a modal)
- Interpret results of simulations Get 3 of 4 questions to level up!

## Addition rule

- Probability with Venn diagrams (Opens a modal)
- Addition rule for probability (Opens a modal)
- Addition rule for probability (basic) (Opens a modal)
- Adding probabilities Get 3 of 4 questions to level up!
- Two-way tables, Venn diagrams, and probability Get 3 of 4 questions to level up!

## Multiplication rule for independent events

- Sample spaces for compound events (Opens a modal)
- Compound probability of independent events (Opens a modal)
- Probability of a compound event (Opens a modal)
- "At least one" probability with coin flipping (Opens a modal)
- Free-throw probability (Opens a modal)
- Three-pointer vs free-throw probability (Opens a modal)
- Probability without equally likely events (Opens a modal)
- Independent events example: test taking (Opens a modal)
- Die rolling probability with independent events (Opens a modal)
- Probabilities involving "at least one" success (Opens a modal)
- Sample spaces for compound events Get 3 of 4 questions to level up!
- Independent probability Get 3 of 4 questions to level up!
- Probabilities of compound events Get 3 of 4 questions to level up!
- Probability of "at least one" success Get 3 of 4 questions to level up!

## Multiplication rule for dependent events

- Dependent probability introduction (Opens a modal)
- Dependent probability: coins (Opens a modal)
- Dependent probability example (Opens a modal)
- Independent & dependent probability (Opens a modal)
- The general multiplication rule (Opens a modal)
- Dependent probability (Opens a modal)
- Dependent probability Get 3 of 4 questions to level up!

## Conditional probability and independence

- Calculating conditional probability (Opens a modal)
- Conditional probability explained visually (Opens a modal)
- Conditional probability using two-way tables (Opens a modal)
- Conditional probability tree diagram example (Opens a modal)
- Tree diagrams and conditional probability (Opens a modal)
- Conditional probability and independence (Opens a modal)
- Analyzing event probability for independence (Opens a modal)
- Calculate conditional probability Get 3 of 4 questions to level up!
- Dependent and independent events Get 3 of 4 questions to level up!

## Math Help: Solving Probability Problems for Students

## Problems With Probability

Solving probability problems can be tricky for many people. Even though the calculations themselves are very simple (basic addition and multiplication), the sequence of math equations is often long and confusing. Let’s break down the problem with a little review. Probability is the chance that an event will occur. Events may be any occurrence such as:

- Going to the doctor
- Standing on a yellow brick or
- Seeing a monkey

## The Coin Toss

When you flip a coin, there are two events can occur:

- The coin could land on heads or
- The coin could land on tails

Since there is no way to land on both sides at the same time, only 1 of these events can occur. Problem 1: So what is the probability of landing on heads ? Heads is 1 out of 2 options. So the probability of getting a heads is 1 out of 2 --the same goes for tails. 1 out of 2 may be written as the ratio (1:2) or a percentage (50%). Slow down! Let’s look at the steps we just completed.

- Count the number of possible events. (2)
- Decide which event you are examining for probability. (Heads)
- Count the number of chances that heads can occur out of the possible events (1)
- Write the number of chances heads could occur over the number of possible events in a ratio. (1:2)

But what does that probability mean? Having a 1:2 probability for heads means that you will get a heads half of the time. This is why coins are used to make decisions, like who goes first in a football game–both teams have a 50% chance of going first and that is fair.

## Want to Test the Probability?

A probability tells you how likely something is to occur. This doesn’t mean that an event is guaranteed to happen, just if it is more or less likely to occur. As we know from the coin model, we have a 50% chance of getting a heads on every toss. This probability doesn’t change no matter how many times we toss the coin. And we can test the probability easily–just toss a coin. Coin Toss Experiment Materials Needed: A paper, a pencil and a quarter

- Toss the quarter 100 times and tally the number of heads and tails.
- Count the number of tallies for each event.
- The number of each event occurring will be very close to 50. (For example, 45 heads and 55 tails)
- If the numbers of heads and tails are NOT close to 50%, toss the quarter 50 more times. The more you toss the quarter, the more likely you are to have an even distribution.

- Count the number of possible events. There are 6 sides to the dice. So there are 6 possible events.
- Decide which event you are examining for probability. The problem let’s us know we are trying to roll a four .
- Count the number of chances that heads can occur out of the possible events. There is only one side of the die that has 4 dots, so there is only 1 chance to roll a four out of 6 total chances.
- Write the number of chances heads could occur over the number of possible events in a ratio. ( 1:6 )

Problem 3 : What is the probability of landing on the black area?

1. Even though there are only 3 different colors, dividing the circle into even sections makes handling the probability easier. This way there are four equally possible outcomes. (white, white, black or orange)

2. We are looking at the probability of landing on black .

3. Black is 1 out of four options.

4. So the probability of landing on black is 1:4 or 25% .

Problem 4: What is the probability of landing on a white area?

- There are four possible options.
- We are figuring out the probability of landing on white .
- Two out of four options are white
- So the probability of landing on white is 1:2 ! (Did you think it was 2:4? Always remember to simplify your ratios!)

## Joint Probability

Here is our final walk-through for solving probability problems. There are many different types of probability that describe the circumstances, or the variables, that impact a certain event. A joint probability is the chance of two events happening back to back. Follow these steps to solve a joint probability.

- Write down the probability of the first event. (Just follow the four-step process we used earlier.)
- Write down the probability of the second event.
- Multiply the two ratios.

Problem 5: What is the probability of tossing two heads in a row? Since we already did the math, we know that the probability of tossing a heads is 1/2 . We also know that this doesn’t change. No matter how many times we flip the coin, there will always be two options, one of which is heads. So the chance of tossing a heads is still 1/2 . (1/2) * (1/2) = ? When multiplying fractions, multiply the numerators (top numbers) and then the denominators (the bottom numbers). Don’t forget to simplify the product.

So the product is 1/4 . There is a 1:4 or 25% chance of getting two heads in a row. Do You Know What That Means? The probability of flipping heads once is greater than the probability of flipping heads twice! When we try to get two events to happen back to back, in a sequence, we lower the probability. Can you guess what happens when we try to get three events to happen? The probability is even lower… Try to figure out the probability of getting three heads in a row. (It’s 1:8!)

## Practice Problems

Practice using the steps to solve the following probability problems. Refer back to the previous examples for help. If you get stuck, take a deep breath and start over with step 1. Keep in mind that these are word problems . List the given and needed information before attempting to solve the problem. That way you won’t miss important details. Problem 6: There are two tokens in a bag, one is white and the other is blue. What is the probability that you will take out the blue token? ( Hint : Coin Toss) Problem 7: Your mother makes turkey on weekdays and beef on the weekends. You don’t know what day it is. What is the probability that you will have turkey for dinner? (Hint: Circle) Problem 8: There are two puppies at Shelly’s house. Shelly hopes that the puppies are girls. What are the chances that one puppy is a girl? What are the chances that both puppies are girls?

## Practice Problem Answers

Answer 6: There are two different events, and you can only have white or blue–just like the coin can only show heads or tails. There is a 1:2 chance that you will draw the blue token. Answer 7: This is a tricky problem because of the wording. You know that your mother makes turkey on 5 days, and beef on 2. Even though there are only two options, it is easier to solve the probability problem by keeping the week divided into equal pieces–7 days. Turkey is dinner 5 out of 7 days. So the probability of having turkey is 5:7 . Answer 8: This was a two-part question. First you want to know what the chances are of one puppy being a girl. Even though there are two puppies, we are only thinking about one right now. That one puppy can either be a boy or a girl. That makes two possible events, with one outcome; so there is a 1:2 chance that one puppy is a girl. The second puppy has the same chance. So using the steps for solving a joint probability there is 1:4 change that both puppies are girls. Another way to look at this is to write out the possible gender combinations.

Four options. “Girl-girl” is 1 out of 4 possible events. Bonus Question: If you know that one puppy is a girl. What is the probability that they are both girls? (The answer is not 1:2)

## Need More Help on How to Solve Probability Problems?

- Ask a teacher. Your teachers are there to help you out. If you don’t understand something in class, raise your hand and ask. If it’s homework that has you stumped, do your best to get it done. Then talk with your teachers the next time you see them. They’ll appreciate the effort, and be more than happy to lend a hand in-class or after-school.
- Work with a friend. Two heads really are better than one. Work through your homework problems with a classmate (unless your teacher asked you not to).
- Call the Homework Hotline . They provide free homework help over the phone. A tutor will listen to your problem and help you find AND UNDERSTAND the right answer. This program is offered through Harvey Mudd College .
- Visit one of the sites below for more examples of probability problems. Sometimes it helps to read two or three different explanations of how to solve a problem. You never know which one is going to “click” and make it clear for you.
- Cut the Knot - The Monty Hall problem is the focus of this page.
- Cut the Knot - The inspiration for the puppy problem.
- Probability Lesson Plan: Teach Probability With Examples - This article is a great advanced resource for teaching and learning probability.
- Photo Credits: “Circle” by Sylvia Cini
- Coins Image by Kevin Schneider from Pixabay
- Dice Image by Erik Stein from Pixabay

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## 15 Probability Questions And Practice Problems for Middle and High School: Harder Exam Style Questions Included

Beki christian.

Probability questions and probability problems require students to work out how likely it is that something is to happen. Probabilities can be described using words or numbers. Probabilities range from 0 to 1 and can be written as fractions, decimals or percentages .

Here you’ll find a selection of probability questions of varying difficulty showing the variety you are likely to encounter in middle school and high school, including several harder exam style questions.

## What are some real life examples of probability?

How to calculate probabilities, probability question: a worked example, 6th grade probability questions, 7th grade probability questions, 8th grade probability questions, 9th grade probability questions, 10th grade probability questions.

- 11th & 12th grade probability questions

## Looking for more middle school and high school probability math questions?

The more likely something is to happen, the higher its probability. We think about probabilities all the time.

For example, you may have seen that there is a 20% chance of rain on a certain day or thought about how likely you are to roll a 6 when playing a game, or to win in a raffle when you buy a ticket.

## 15 Probability Questions Worksheet

Want the 15 multiple choice probability questions from this blog in a handy downloadable format? Then look no further!

The probability of something happening is given by:

We can also use the following formula to help us calculate probabilities and solve problems:

- Probability of something not occuring = 1 – probability of if occurring P(not\;A) = 1 - P(A)
- For mutually exclusive events: Probability of event A OR event B occurring = Probability of event A + Probability of event B P(A\;or\;B) = P(A)+P(B)
- For independent events: Probability of event A AND event B occurring = Probability of event A times probability of event B P(A\;and\;B) = P(A) × P(B)

Question: What is the probability of getting heads three times in a row when flipping a coin?

When flipping a coin, there are two possible outcomes – heads or tails. Each of these options has the same probability of occurring during each flip. The probability of either heads or tails on a single coin flip is ½.

Since there are only two possible outcomes and they have the same probability of occurring, this is called a binomial distribution.

Let’s look at the possible outcomes if we flipped a coin three times.

Let H=heads and T=tails.

The possible outcomes are: HHH, THH, THT, HTT, HHT, HTH, TTH, TTT

Each of these outcomes has a probability of ⅛.

Therefore, the probability of flipping a coin three times in a row and having it land on heads all three times is ⅛.

## Middle school probability questions

In middle school, probability questions introduce the idea of the probability scale and the fact that probabilities sum to one. We look at theoretical and experimental probability as well as learning about sample space diagrams and venn diagrams.

1. Which number could be added to this spinner to make it more likely that the spinner will land on an odd number than a prime number?

Currently there are two odd numbers and two prime numbers so the chances of landing on an odd number or a prime number are the same. By adding 3, 5 or 11 you would be adding one prime number and one odd number so the chances would remain equal.

By adding 9 you would be adding an odd number but not a prime number. There would be three odd numbers and two prime numbers so the spinner would be more likely to land on an odd number than a prime number.

2. Ifan rolls a fair dice, with sides labeled A, B, C, D, E and F. What is the probability that the dice lands on a vowel?

A and E are vowels so there are 2 outcomes that are vowels out of 6 outcomes altogether.

Therefore the probability is \frac{2}{6} which can be simplified to \frac{1}{3} .

3. Max tested a coin to see whether it was fair. The table shows the results of his coin toss experiment:

Heads Tails

26 41

What is the relative frequency of the coin landing on heads?

Max tossed the coin 67 times and it landed on heads 26 times.

\text{Relative frequency (experimental probability) } = \frac{\text{number of successful trials}}{\text{total number of trials}} = \frac{26}{67}

4. Grace rolled two dice. She then did something with the two numbers shown. Here is a sample space diagram showing all the possible outcomes:

What did Grace do with the two numbers shown on the dice?

Add them together

Subtract the number on dice 2 from the number on dice 1

Multiply them

Subtract the smaller number from the bigger number

For each pair of numbers, Grace subtracted the smaller number from the bigger number.

For example, if she rolled a 2 and a 5, she did 5 − 2 = 3.

5. Alice has some red balls and some black balls in a bag. Altogether she has 25 balls. Alice picks one ball from the bag. The probability that Alice picks a red ball is x and the probability that Alice picks a black ball is 4x. Work out how many black balls are in the bag.

Since the probability of mutually exclusive events add to 1:

\begin{aligned} x+4x&=1\\\\ 5x&=1\\\\ x&=\frac{1}{5} \end{aligned}

\frac{1}{5} of the balls are red and \frac{4}{5} of the balls are blue.

6. Arthur asked the students in his class whether they like math and whether they like science. He recorded his results in the venn diagram below.

How many students don’t like science?

We need to look at the numbers that are not in the ‘Like science’ circle. In this case it is 9 + 7 = 16.

## High school probability questions

In high school, probability questions involve more problem solving to make predictions about the probability of an event. We also learn about probability tree diagrams, which can be used to represent multiple events, and conditional probability.

7. A restaurant offers the following options:

Starter – soup or salad

Main – chicken, fish or vegetarian

Dessert – ice cream or cake

How many possible different combinations of starter, main and dessert are there?

The number of different combinations is 2 × 3 × 2 = 12.

8. There are 18 girls and 12 boys in a class. \frac{2}{9} of the girls and \frac{1}{4} of the boys walk to school. One of the students who walks to school is chosen at random. Find the probability that the student is a boy.

First we need to work out how many students walk to school:

\frac{2}{9} \text{ of } 18 = 4

\frac{1}{4} \text{ of } 12 = 3

7 students walk to school. 4 are girls and 3 are boys. So the probability the student is a boy is \frac{3}{7} .

9. Rachel flips a biased coin. The probability that she gets two heads is 0.16. What is the probability that she gets two tails?

We have been given the probability of getting two heads. We need to calculate the probability of getting a head on each flip.

Let’s call the probability of getting a head p.

The probability p, of getting a head AND getting another head is 0.16.

Therefore to find p:

The probability of getting a head is 0.4 so the probability of getting a tail is 0.6.

The probability of getting two tails is 0.6 × 0.6 = 0.36 .

10. I have a big tub of jelly beans. The probability of picking each different color of jelly bean is shown below:

If I were to pick 60 jelly beans from the tub, how many orange jelly beans would I expect to pick?

First we need to calculate the probability of picking an orange. Probabilities sum to 1 so 1 − (0.2 + 0.15 + 0.1 + 0.3) = 0.25.

The probability of picking an orange is 0.25.

The number of times I would expect to pick an orange jelly bean is 0.25 × 60 = 15 .

11. Dexter runs a game at a fair. To play the game, you must roll a dice and pick a card from a deck of cards.

To win the game you must roll an odd number and pick a picture card. The game can be represented by the tree diagram below.

Dexter charges players $1 to play and gives $3 to any winners. If 260 people play the game, how much profit would Dexter expect to make?

Completing the tree diagram:

Probability of winning is \frac{1}{2} \times \frac{4}{13} = \frac{4}{26}

If 260 play the game, Dexter would receive $260.

The expected number of winners would be \frac{4}{26} \times 260 = 40

Dexter would need to give away 40 × $3 = $120 .

Therefore Dexter’s profit would be $260 − $120 = $140.

12. A fair coin is tossed three times. Work out the probability of getting two heads and one tail.

There are three ways of getting two heads and one tail: HHT, HTH or THH.

The probability of each is \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}

Therefore the total probability is \frac{1}{8} +\frac{1}{8} + \frac{1}{8} = \frac{3}{8}

## 11th/12th grade probability questions

13. 200 people were asked about which athletic event they thought was the most exciting to watch. The results are shown in the table below.

A person is chosen at random. Given that that person chose 100m, what is the probability that the person was female?

Since we know that the person chose 100m, we need to include the people in that column only.

In total 88 people chose 100m so the probability the person was female is \frac{32}{88} .

14. Sam asked 50 people whether they like vegetable pizza or pepperoni pizza.

37 people like vegetable pizza.

25 people like both.

3 people like neither.

Sam picked one of the 50 people at random. Given that the person he chose likes pepperoni pizza, find the probability that they don’t like vegetable pizza.

We need to draw a venn diagram to work this out.

We start by putting the 25 who like both in the middle section. The 37 people who like vegetable pizza includes the 25 who like both, so 12 more people must like vegetable pizza. 3 don’t like either. We have 50 – 12 – 25 – 3 = 10 people left so this is the number that must like only pepperoni.

There are 35 people altogether who like pepperoni pizza. Of these, 10 do not like vegetable pizza. The probability is \frac{10}{35} .

15. There are 12 marbles in a bag. There are n red marbles and the rest are blue marbles. Nico takes 2 marbles from the bag. Write an expression involving n for the probability that Nico takes one red marble and one blue marble.

We need to think about this using a tree diagram. If there are 12 marbles altogether and n are red then 12-n are blue.

To get one red and one blue, Nico could choose red then blue or blue then red so the probability is:

- Ratio questions
- Algebra questions
- Trigonometry questions
- Venn diagram questions
- Long division questions
- Pythagorean theorem questions

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The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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The application or uses of probability can be seen in quantitative aptitude as well as in daily life. It is needful to learn the basic concept of probability. We will cover the basics as well as the hard level problems for all levels of students for all competitive exams especially SBI PO, SBI CLERK, IBPS PO, IBPS CLERK, RRB PO, NICL AO, LIC AAO, SNAP, MAT, SSC CGL etc.

Definition: Probability means the possibility or chances of an event occurring or happening. For example, when a coin is tossed, then we will get ahead or tail. It is a state of probability.

In an event, the happening probability is equal to the ratio of favourable outcomes to the total number of possible outcomes. It represents as,

Number of favourable outcomes = ____________________________________ Total number of possible outcomes

Sample Space:-

It is a set of all possible outcomes of an experiment. It is denoted by S. For example, the Sample space of a die, S = [ 1, 2, 3 , 4, 5, 6] The Sample space of a coin, S= [ Head, Tail]

Types of questions asked in the competitive exam:

1) Based on Coins

2) Based on Dice

3) Based on playing Cards

4) Based on Marbles or balls

5) Miscellaneous

Important Questions:

1. Question A coin is thrown two times .what is the probability that at least one tail is obtained?

A) 3/4 B) 1/4 C) 1/3 D) 2/3 E) None of these

Answer :- A Sol:

Sample space = [TT, TH, HT,HH] Total number of ways = 2 × 2 = 4. Favourite Cases = 3 P (A) = 3/4

Tricks:- P (of getting at least one tail) = 1 – P (no head)⇒ 1 – 1/4 = 3/4

2. Question What is the probability of getting a numbered card when drawn from the pack of 52 cards?

A) 1/13 B) 1/9 C) 9/13 D) 11/13 E) None of these

Answer :- C Sol: Total Cards = 52. Numbered Cards = 9 (2,3,4,5,6,7,8,9,10) in each suit Numbered cards in four suit = 4 ×9 = 36 P (E) = 36/52 = 9/13

3.Question There are 7 purple clips and 5 brown clips. Two clips are selected one by one without replacement. Find the probability that the first is brown and the second is purple.

A) 1/35 B) 35/132 C) 1/132 D) 35/144 E) None of these

Answer :- B Sol:

P (B) × P (P) = (5/12) x (7/11) = 35/132

4.Question Find the probability of getting a sum of 8 when two dice are thrown?

A) 1/8 B) 1/5 C) 1/4 D) 5/36 E) 1/3

Answer 😀 Sol: Total number of ways = 6 × 6 = 36 ways. Favorable cases = (2 , 6) (6, 2) (3, 5) (5, 3) (4, 4) — 5 ways. P (A) = 5/36 = 5/36

5.Question Find the probability of an honour card when a card is drawn at random from the pack of 52 cards.

A) 4/13 B) 1/3 C) 5/12 D) 7/52 E) None of these

Answer :-A Sol: Honor cards = 4 (A, J, Q, K) in each suit Honor cards in 4 suit = 4 × 4 = 16 P (honor card) = 16/52 = 4/13

6. Question What is the probability of a face card when a card is drawn at random from the pack of 52 cards?

A) 1/13 B) 2/13 C) 3/13 D) 4/13 E) 5/13

Answer :-C Solution: face cards = 3 (J,Q,K) in each suit Face cards in 4 suits = 3 × 4 = 12 Cards. P (face Card) = 12/52 = 3/13

7.Question If two dice are rolled together then find the probability as getting at least one ‘3’?

A) 11/36 B) 1/12 C) 1/36 D) 13/25 E) 13/36

Answer :- A Sol: Total number of ways = 6 × 6 = 36. Probability of getting number ‘3′ at least one time = 1 – (Probability of getting no number 4) = 1 – (5/6) x (5/6) = 1 – 25/36 = 11/36

8. Question If a single six-sided die is rolled then find the probability of getting either 3 or 4.

A) 1/2 B) 1/3 C) 1/4 D) 2/3 E) 1/6

Answer:- B Solution:- Total outcomes = 6 Probability of getting a single number when rolled a die = 1/6 So, P(3) = 1/6 and P(4) = 1/6 Thus, the probability of getting either 3 or 4 = P(3)+P(4) = 1/6 + 1/6 = 1/3

9. Question A container contains 1 red, 3 black, 2 pink and 4 violet gems. If a single gem is chosen at random from the container, then find the probability that it is violet or black?

A) 1/10 B) 3/10 C) 7/10 D) 9/10 E) None of these

Answer :-C Sol :- Total gems =( 1 + 3 + 2 + 4 ) = 10 probability of getting a violet gem = 4/10 Probability of getting a black gem = 3/10 Now, P ( Violet or Black) = P(violet) + P(Black) = 4/10 + 3/10 = 7/10

10.Question A jar contains 63 balls ( 1,2,3,……., 63). Two balls are picked at random from the jar one after one and without any replacement. what is the probability that the sum of both balls drawn is even?

A) 5/21 B) 3/23 C) 5/63 D) 19/63 E) None of these

Answer :- E Sol. Sum of numbers can be even if we add either two even numbers or two odd numbers.

Number of even numbers from 1 to 63 = 31

Number of odd numbers from 1 to 63 = 32

Probability of getting two even numbers = (32/63) * (31/62) = 16/63

Probability of getting two odd numbers = (31/63) * (30/62) = 5/21

P(two even numbers OR two odd numbers) = 16/63 + 5/21 = 31/63

11.Question There are 30 students in a class, 15 are boys and 15 are girls. In the final exam, 5 boys and 4 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an ‘A grade student?

A) 1/4 B) 3/10 C) 1/3 D) 2/3 E) None of these

Answer:- D Sol:

Here, the total number of boys = 15 and the total number of girls = 15

Also, girls getting A grade = 4 and boys getting an A grade = 5 Probability of choosing a girl = 15/30

Probability of choosing A grade student= 9/30

Now, an A-grade student chosen can be a girl. So the probability of choosing it = 4/30

Required probability of choosing a girl or an A grade student = 15/30 + 9/30 – 4/30 = 1/2 + 3/10 – 2/15 = 2/3 12. Question What is the probability when a card is drawn at random from a deck of 52 cards is either an ace or a club?

A) 2/13 B) 3/13 C) 4/13 D) 5/23 E) None of these

Answer:- C Sol: There are 4 aces in a pack, 13 club cards and 1 ace of club card.

Now, the probability of getting an ace = 4/52

Probability of getting a club = 13/52

Probability of getting an ace of club = 1/52

Required probability of getting an ace or a club

= 4/52 + 13/52 – 1/52 = 16/52 = 4/13

13. Question One card is drawn from a deck of 52 cards well shuffling. Calculate the probability that the card will not be a king.

A) 12/13 B) 3/13 C) 7/13 D) 5/23 E) None of these

Solution:

Well-shuffling ensures equally likely outcomes. Total king of a deck = 4

The number of favourable outcomes F= 52 – 4 = 48

The number of possible outcomes = 52

Therefore, the required probability

= 48/52 = 12/13

14.Question If P(A) = 7/13, P(B) = 9/13 and P(A∩B) = 4/13, find the value of P(A|B).

A) 1/9 B) 2/9 C) 3/9 D) 4/9 E) None of these

Answer :- D Solution:

P(A|B) = P(A∩B)/P(B) = (4/13)/(9/13) = 4/9.

15. Question A one rupee coin and a two rupee coin are tossed once, then calculate a sample space.

A) [ HH, HT, TH, TT]

B) [ HH, TT]

C) [ TH, HT]

D) [HH, TH, TT]

E) None of these

The outcomes are either Head (H) or tail(T).

Now,heads on both coins = (H,H) = HH

Tails on both coins = ( T, T) = TT

Probability of head on one rupee coin and Tail on the two rupee coins = (H, T) = HT

And Tail on one rupee coin and Head on the two rupee coin = (T, H) = TH

Thus, the sample space ,S = [HH, HT, TH, TT]

16. Question There are 20 tickets numbered 1 to 20. These tickets are mixed up and then a ticket is drawn at random. Find the probability that the ticket drawn has a number which is a multiple of 4 or 5?

A) 1/4 B) 2/13 C) 8/15 D) 9/20 E) None of these

Here, S = {1, 2, 3, 4, …., 19, 20} = 20

Multiples of 4: 4, 8, 12, 16, 20 (5 tickets) Multiples of 5: 5, 10, 15, 20 (4 tickets)

Notice that ticket number 20 is a multiple of both 4 and 5, so we have counted it twice. Therefore, we need to subtract one from the total count.

Total number of tickets with numbers that are multiples of 4 or 5: 5 + 4 – 1 = 8

The total number of tickets is 20, so the probability of drawing a ticket with a number that is a multiple of 4 or 5 is:

P = 8/20 = 2/5 = 0.4

Therefore, the probability that the ticket drawn has a number which is a multiple of 4 or 5 is 0.4 or 40%.

Direction ( 17 – 19):- In a school the total number of students is 300, 95 students like chicken only, 120 students like fish only, 80 students like mutton only and 5 students do not like anything above. If randomly one student is chosen, find the probability that

17) The student likes mutton.

18 ) he likes either chicken or mutton

19 ) he likes neither fish nor mutton.

Solution( 17-19):-

The total number of favourable outcomes = 300 (Since there are 300 students altogether).

The number of times a chicken liker is chosen = 95 (Since 95 students like chicken).

The number of times a fish liker is chosen = 120.

The number of times a mutton liker is chosen = 80.

The number of times a student is chosen who likes none of these = 5.

17. Question Find the probability that the student like mutton?

A) 3/10 B) 4/15 C) 1/10 D) 1/15 E) None of these

Answer:- B Solution:-

Therefore, the probability of getting a student who likes mutton

= 80/300 = 4/15

18. Question What is the probability that the student likes either chicken or mutton?

A) 7/12 B) 5/12 C) 3/4 D) 1/12 E) None of these

Answer:- A Solution:-

The probability of getting a student who likes either chicken or mutton = (95+80)/300 = 175/300 = 7/12

19. Question Find the probability that the student likes neither fish nor mutton.

A) 1/2 B) 1/5 C) 1/3 D) 1/4 E) 1/6

Answer:- C Solution:- The probability of getting a student who likes neither fish nor mutton = (300–120−80)/300 = 100/300 = 1/3

Direction ( 20-22):- A box contains 90 number plates numbered 1 to 90. If one number plate is drawn at random from the box then find out the probability that

20) The number is a two-digit number

21) The number is a perfect square

22) The number is a multiply of 5

20. Question Find the probability that the number is a two-digit number.

A) 1/9 B) 1/10 C) 9/10 D) 7/10 E) None of these

Answer:-C Solution : Total possible outcomes = 90 (Since the number plates are numbered from 1 to 90).

Number of favourable outcomes = 90 – 9 = 81 ( here, except 1 to 9, other numbers are two-digit number.)

Thus required probability = Number of Favourable Outcomes /Total Number of Possible Outcomes = 81/90 = 9/10.

21. Question What is the probability that the number is a perfect square?

A) 1/9 B) 1/10 C) 9/10 D) 1/7 E) None of these

Answer:- B Solution:- Total possible outcomes = 90. Number of favourable outcomes = 9 [here 1, 4, 9, 16, 25, 36, 49, 64 and 81 are the perfect squares] Thus the required probability = 9/90 =1/10

22.Question Find the probability that the number is a multiple of 5.

A) 1/5 B) 1/6 C) 1/10 D) 1/8 E) 9/10

Answer:- A Solution:- Total possible outcomes = 90. Number of favourable outcomes = 18 (here, 5 × 1, 5 × 2, 5 × 3, …., 5 × 18 are multiple of 5).

Thus, the required probability= 18/90 =1/5

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## Perplexing Probability

Flex your probability skills with some surprisingly counter-intuitive games and puzzles.

Exposing Misconceptions

Challenge yourself with probability puzzles that trip up plenty of experts.

Gaming Strategies

Learn some quick tips for solving probability problems.

Probability is Everywhere

Discover some counterintuitive results when you applying probability to these real-life scenarios.

## End of Unit 1

Complete all lessons above to reach this milestone.

0 of 3 lessons complete

- The Monty Hall Game

The famous mind-bender: find the car and avoid the man-eating goat!

More Man-Eating Goats

What happens when Monty Hall has a lot more doors?

The Boy-or-Girl Paradox

A small change of conditions can make a big difference.

Tuesday Changes Everything

Explore this counterintuitive twist to the Boy-or-Girl Paradox.

Bayes' Theorem Magic

This formula explains the behavior of many dependent probability scenarios.

## End of Unit 2

0 of 5 lessons complete

Crazy Dice Warmup

See what happens when the dice don't read 1 to 6 anymore.

Competitive Design

Design your own die and try to beat your friends.

Beating a Standard Die (I)

Under what conditions, can you beat a standard die?

Beating a Standard Die (II)

Explore more scenarios that match a custom die against a standard one.

3-Player Competitions

Add a third player for added perplexity.

Non-Transitive Superiority

When A beats B and B beats C, does A always beat C?

## End of Unit 3

0 of 6 lessons complete

## Course description

Probability is full of counter-intuitive results and paradoxes. Challenge yourself by learning how to mathematically master some very surprising games and experiments including the Monty Hall game show, the Boy or Girl paradox, and the Tuesday paradox. Some prior knowledge of probability at the level of the Probability Fundamentals course is useful but not required for tackling this course.

## Topics covered

- Bayes' Theorem
- Complementary Probability
- Conditional Probability
- Expected Value
- Probability Distributions
- Probability Misconceptions
- Russian Roulette Variants
- The Boy or Girl Paradox
- The Tuesday Paradox

## Prerequisites

- Reasoning with Algebra
- Applied Probability
- Statistics Fundamentals

- Basic Tricks
- Number Series
- Aptitude Tricks
- Reasoning Tricks
- Roman Numeral
- Math Formulas

## Probability Tricks

Before doing anything we recommend you to do a math practice set. Then find out twenty math problems related to this topic and write those on a paper. Do first ten maths using basic formula of this math topic. You also need to keep track of the time. Write down the time taken by you to solve those questions. Now read our examples on probability shortcut tricks and practice few questions. After finishing this do remaining questions using Probability shortcut tricks. Again keep track of the time. The timing will be surely improved this time. But this is not all you want. You need more practice to improve your timing more.

## Few Important things to Remember

You all know that math portion is very much important in competitive exams. That doesn’t mean that other topics are less important. You can get a good score only if you get a good score in math section. A good score comes with practice and practice. All you need to do is to do math problems correctly within time, and only shortcut tricks can give you that success. But it doesn’t mean that without using shortcut tricks you can’t do any math problems. You may have that potential that you may do maths within time without using any shortcut tricks.

But, so many people can’t do this. So Probability shortcut tricks here for those people. We try our level best to put together all types of shortcut methods here. But it possible we miss any. We appreciate if you share that with us. Your help will help others.

Probability is very important chapter,Here we put down some formula and Facts that are based on this chapter after that we give some examples related on formula and chapter,which help you better understand about this chapter. In exams only Dice, Coin and ball related examples are given.

Now we will discuss some basic ideas of Probability . On the basis of these ideas we will learn trick and tips of shortcut probability. If you think that how to solve probability questions using probability shortcut tricks , then further studies will help you to do so.

## Important Facts

- When we throw a coin on the air, the coin appears either a Head (H) or a Tail (T).
- A dice is a solid cube having 6 faces, marked as 1, 2, 3, 4, 5, 6 gradually. When we throw a dice the outcome is the number that appears on it’s upper face of the dice.

## What is Experiment?

The actions of that Appear some well-defined outcomes is called an Experiment.

## What is Random Experiment?

A Random experiment is an experiment in which all probable outcomes are known and we cannot predicted the accurate outcome in advances.

## What is Sample Space?

Sample Space is an set S of all possible outcomes in a particular actions.

Examples of Sample Spaces

- In tossing a coin on air, S = { H, T }.
- If two coin tossed, then S = { HH, HT, TH, TT }.
- For rolling a dice, S = { 1, 2, 3, 4, 5, 6 }.

## What is an Event?

Any subset of a sample space is called an Event.

Probability of occurrence of an Event Probability of any Event P(E) = n(E) / n(S).

Here, P(E) = Probability of Event. n(E) = Total number of required outcomes. n(S) = Total number of Possible outcomes.

Result on Probability

- P(S) = 1 ( 1 means maximum probability is always 1 ).
- P(∅) = 0 ( Maximum probability is always 0 ).
- 0 < P(E) < 1
- For any event X and Y, we have, P(X ∪ Y) = P (X) + (Y) – P (X ∩ Y).

## Few examples of Probability with Shortcut Tricks

- 1. Probability Problem on Coin
- 2. Probability problem on Balls
- 3. Probability problem on Dice
- 4. Probability Example 1
- 5. Probability Example 2
- 6. Probability Example 3
- 7. Probability Example 4

We provide few tricks on Probability Tricks. Please visit this page to get updates on more Math Shortcut Tricks . You can also like our facebook page to get updates.

If you have any question regarding Probability Tricks then please do comment on below section. You can also send us message on facebook .

Quite useful actually… Thanks

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Thank you very much

A bag contains 4 red balls and 6 yellow balls. If 2 are drawn from the bag randomly. What’s the probability that one ball is red?

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## An Explosive Hearing in Trump’s Georgia Election Case

Fani t. willis, the district attorney, defended her personal conduct as defense lawyers sought to disqualify her from the prosecution..

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In tense proceedings in Georgia, a judge will decide whether Fani T. Willis, the Fulton County district attorney, and her office should be disqualified from their prosecution of former President Donald J. Trump.

Richard Fausset, a national reporter for The Times, talks through the dramatic opening day of testimony, in which a trip to Belize, a tattoo parlor and Grey Goose vodka all featured.

## On today’s episode

Richard Fausset , a national reporter for The New York Times.

## Background reading

With everything on the line, Ms. Willis delivered raw testimony .

What happens if Fani Willis is disqualified from the Trump case?

Read takeaways from the hearing .

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The easiest way to solve these types of probability problems is to write out all the possible dice combinations (that's called writing a sample space ). A very simple example, if you want to know the probability of rolling a double with two die, your sample space would be: [1] [1], [1] [2], [1] [3], [1] [4], [1] [5], [1] [6],

To solve a probability problem identify the event, find the number of outcomes of the event, then use probability law: \ (\frac {number\ of \ favorable \ outcome} {total \ number \ of \ possible \ outcomes}\) Probability Problems The Absolute Best Books to Ace Pre-Algebra to Algebra II The Ultimate Algebra Bundle From Pre-Algebra to Algebra II

Tips The probability of an event can only be between 0 and 1 and can also be written as a percentage. The probability of event A is often written as P ( A) . If P ( A) > P ( B) , then event A has a higher chance of occurring than event B . If P ( A) = P ( B)

Solution: The only way to obtain a sum of 10 from two 5-sided dice is that both die shows 5 face up. Therefore, the probability is simply \frac15 \times \frac15 = \frac1 {25} = .04 51 × 51 = 251 =.04 \dfrac {1} {4} 41 \dfrac {1} {32} 321 \dfrac {32} {13} 1332 \dfrac {13} {32} 3213

Types of Experiment: While studying probability theory, we will frequently use the term 'experiment' which means an operation which can produce well defined outcome (s). There are two types of experiments: (i) Deterministic Experiment: The experiments whose outcome is same when done under exact conditions are called Deterministic experiment.

Practice test AP formulas Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Statistics Probability Bayes rule Combinations & permutations Factorial Bartlett's test Event counter

1. Simulation We can use a simulation to estimate a probability by doing many trials of simulating a random phenomena. We know that as the number of trials increases, the proportion of times that our chosen event occurs will approach the true probability (Law of Large Numbers!).

Probability is the likelihood or chance of an event occurring. Probability = the number of ways of achieving success. the total number of possible outcomes.

We will now look at some examples of probability problems. Example: At a car park there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are lorries. If every vehicle is equally likely to leave, find the probability of: a) a van leaving first. b) a lorry leaving first.

Here are some helpful study tips to help you get well-prepared for a probability exam. Principles of probability The mathematics field of probability has its own rules, definitions, and laws, which you can use to find the probability of outcomes, events, or combinations of outcomes and events.

1 Choose an event with mutually exclusive outcomes. Probability can only be calculated when the event whose probability you're calculating either happens or doesn't happen. The event and its opposite both cannot occur at the same time. Rolling a 5 on a die, a certain horse winning a race, are examples of mutually exclusive events.

1 Understand the formula for determining probability. Probability is the likelihood of a random event happening. [1] It is usually expressed as a ratio.

Conclusion Get the keyword. This is one of the important tips to solve the probability term problem that involves getting the keyword. This will help the learners to recognize which theorem is used for solving the probability problems. The keywords can be "or" "and" and "not."

How to Solve Probability Word Problems | P (A and B) | P (A or B) | Binomial Probability - YouTube © 2024 Google LLC...

Statistics and probability 16 units · 157 skills. Unit 1 Analyzing categorical data. Unit 2 Displaying and comparing quantitative data. Unit 3 Summarizing quantitative data. Unit 4 Modeling data distributions. Unit 5 Exploring bivariate numerical data. Unit 6 Study design. Unit 7 Probability. Unit 8 Counting, permutations, and combinations.

Problem 1: So what is the probability of landing on heads? Heads is 1 out of 2 options. So the probability of getting a heads is 1 out of 2 --the same goes for tails. 1 out of 2 may be written as the ratio (1:2) or a percentage (50%). Slow down!

We can also use the following formula to help us calculate probabilities and solve problems: Probability of something not occuring = 1 - probability of if occurring P (notA) = 1 − P (A) P (not A) = 1 − P (A) For mutually exclusive events: Probability of event A OR event B occurring = Probability of event A + Probability of event B

1. Question A coin is thrown two times .what is the probability that at least one tail is obtained? A) 3/4 B) 1/4 C) 1/3 D) 2/3 E) None of these Answer :- A Sol: Sample space = [TT, TH, HT,HH] Total number of ways = 2 × 2 = 4.

Packed with practical tips and techniques for solving probability problems Increase your chances of acing that probability exam -- or winning at the casino! Whether you're hitting the books for a probability or statistics course or hitting the tables at a casino, working out probabilities can be problematic. This book helps you even the odds. Using easy-to-understand explanations and examples ...

Probability is full of counter-intuitive results and paradoxes. Challenge yourself by learning how to mathematically master some very surprising games and experiments including the Monty Hall game show, the Boy or Girl paradox, and the Tuesday paradox. Some prior knowledge of probability at the level of the Probability Fundamentals course is useful but not required for tackling this course.

. All probability tricks are provided here. Visitors are requested to carefully read all shortcut examples. These examples will help you to understand shortcut tricks on Probability. Before doing anything we recommend you to do a math practice set. Then find out twenty math problems related to this topic and write those on a paper.

The examples cover variety of probability problems. Suggested Action. FREE Live Master Classes by our Star Faculty with 20+ years of experience. Register Now . ... Sol: Probability of the problem getting solved = 1 - (Probability of none of them solving the problem) Probability of problem getting solved = 1 - (5/7) x (3/7) x (5/9) = (122/147)

Solve probability word problems step by step. probability-problems-calculator. en. Related Symbolab blog posts. Middle School Math Solutions - Simultaneous Equations Calculator. Solving simultaneous equations is one small algebra step further on from simple equations. Symbolab math solutions...

Fani T. Willis, the district attorney, defended her personal conduct as defense lawyers sought to disqualify her from the prosecution.