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By the end of this lesson, students will be able to calculate theoretical and experimental probability using appropriate formulas and interpret the results in real-world contexts.

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Students will be able to: 1) Define probability and identify sample spaces, 2) Calculate theoretical probability using P(event) = favorable outcomes/total outcomes, 3) Conduct experiments to determine experimental probability, 4) Compare theoretical vs experimental probability results.

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How can we predict the likelihood of events occurring, and why might our predictions differ from actual results? This question will guide our exploration of probability concepts throughout the lesson.

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Introduction to key vocabulary: probability, outcome, event, sample space, theoretical probability, experimental probability. Formula: P(event) = Number of favorable outcomes / Total number of possible outcomes. Examples using dice, coins, and cards.

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Guided Practice: Coin Flip Investigation

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Guided Practice: Coin Flip Investigation

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Students work in pairs to flip a coin 20 times, record results, and calculate experimental probability. Compare results with theoretical probability of 1/2. Discuss why results may vary between groups.

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Coin Toss Probability Investigation Worksheet

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Flipping a coin is all fun, but when combined with mathematical thinking, it becomes a powerful tool for understanding probability. Our Coin Toss Probability Investigation Worksheet transforms this simple action into an engaging exploration of chance and likelihood. Key features of this investigation include: Step-by-step recording of coin flip results Data analysis opportunities Real-world connections to probability Visual representations of outcomes The hands-on investigation engages students in a comprehensive exploration of probability concepts. They collect and analyze real data from their coin tosses, learning to calculate experimental probability based on their results. As they work through the activity, students compare their actual results with theoretical probabilities, developing crucial prediction skills. This practical experience helps them understand the concept of fairness in chance events, bridging the gap between mathematical theory and real-world application. Perfect for mathematics classes, this worksheet bridges the gap between abstract probability concepts and concrete understanding. Students move beyond memorizing that a coin has a “50-50 chance” to experiencing and analyzing probability in action.

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Collaborative Activity: Probability with Playing Cards

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Collaborative Activity: Probability with Playing Cards

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Small groups use a standard deck of cards to calculate probabilities of drawing different suits, face cards, or specific numbers. Students can use Padlet to share their findings and compare strategies with other groups.

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A hypergeometric calculator to calculate card draw probabilities. This hypergeometric calculator is perfect for MTG (Magic: The Gathering), Yu-Gi-Oh!, Marvel Snap, Hearthstone, Texas Hold’em and other card games.

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Students solve probability problems involving weather forecasts, sports statistics, and gaming scenarios. Problems range from basic calculations to compound events for differentiated learning.

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For students needing additional support: Use tree diagrams, tables, and visual fraction models to represent probability situations. Start with simple two-outcome events before progressing to more complex scenarios.

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Advanced students explore compound events using 'AND' and 'OR' scenarios. Calculate probabilities of multiple independent events occurring together or separately using multiplication and addition rules.

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Request a demo

Here you will learn about compound probability for both independent and dependent variables and also how to use sample space diagrams and Venn diagrams for compound probabilities.

Students will first learn about compound probability as part of statistics and probability in 7 th grade and continue to learn about compound probability in high school.

What is compound probability?

Compound probability is how likely two or more events are to occur at the same time or in succession. To calculate the likelihood of a certain combination of outcomes, you can use the fundamental counting principle or represent the sample space with a diagram.

The fundamental counting principle says that if one event has m outcomes and a second independent event has n outcomes, then the total number of combined outcomes is m \times n.

For example,

A six-sided die is rolled and the suit of a card randomly chosen from a deck of cards is noted.

The number of different outcomes can be calculated by thinking about how many different options there are for each suit of a card.

Each suit of a card could be paired with 6 different numbers.

Since there are 4 suits, this means there will be 4 groups of 6 pairings.

4\times 6=24, so there are 24 different combinations of the numbers 1-6 and the 4 different suits of a card.

Step-by-step guide: Fundamental counting principle

Now let's look at the same example, but in a sample space diagram.

Each row represents a different number rolled on the die. Each column represents a different suit of a card. There are 24 different outcomes for these combined events.

The probability of rolling a two and drawing a heart from the deck is \cfrac{1}{24} because this is one combination out of twenty-four possible combinations.

Step-by-step guide: Sample Space

For an overview of the rules for calculating compound probabilities, see also: How to calculate probability.

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Common Core State Standards

How does this relate to 7 th grade math?

Grade 7 - Statistics & Probability (7.SP.C.8)
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (example, "rolling double sixes"), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

How to calculate the total possible outcomes of compound events

In order to calculate the total possible outcomes of compound events:

1. Identify the number of sets to be selected from.

2. Identify the number of items to select from each set.

3. Multiply the number of items in each set. If selecting two items from a set, calculate \textbf{n}\times \left( \textbf{n}-\bf{1} \right) or \cfrac{\textbf{n}\times \left( \textbf{n}-\bf{1} \right)}{\textbf{2}} if order doesn't matter.

Compound probability examples

Example 1: compound probability of independent events

Lily can choose a sandwich and a dessert as part of her lunch combo. There are 5 types of sandwiches and 8 different desserts. How many possible combinations can Lily select?

Identify the number of sets to be selected from.

There are two sets to select from, sandwiches and desserts.

2 Identify the number of items to select from each set.

There are 5 types of sandwiches and 8 different desserts.

3 Multiply the number of items in each set. If selecting two items from a set, calculate \textbf{n}\times \left( \textbf{n}-\bf{1} \right) or \cfrac{\textbf{n}\times \left( \textbf{n}-\bf{1} \right)}{\textbf{2}} if order doesn't matter.

Using the counting principle, you multiply

5 \times 8=40

There are 40 total combinations that Lily can select.

Example 2: compound probability of independent events

Tyrese is shopping for clothes and can pick a T-shirt, a pair of sweatpants, and a hoodie as part of a mix-and-match deal. There are 5 different T-shirt colors, 6 sweatpants colors, and 2 hoodie colors. How many possible combinations can Tyrese select?

5 \times 6 \times 2=60

Example 3: selecting a pair from one set when order matters

A manager is selecting two employees to present at a company meeting, deciding who will present first and who will go second. If there are 15 employees to choose from, how many different ways can the manager make this selection?

15 \times 14=210

How to use a sample space

In order to find a probability using a sample space diagram:

1. Use information provided to decide whether to write a list or create a table to find all possible outcomes.

2. Systematically write the list or fill in the table by either listing outcomes or performing operations with values.

3. Use the information from the list or table to find any probabilities required.

Example 4: writing a list of possible outcomes

A fair coin is flipped and the spinner is spun.

Write a list of all possible combinations of outcomes and find the probability of getting heads and spinning yellow.

P(\text { heads and yellow })=\cfrac{1}{6}.

Example 5: creating a tree diagram of possible outcomes

Darshan will randomly be placed into 1 art class and 1 foreign language class. The options for art classes are: painting, theater, chorus and band. The options for foreign language classes are: French, Mandarin and Spanish.

Write a list of all possible combinations of outcomes and find the probability of Darshan being placed into chorus and French.

Example 6: creating a sample space table

The spinner is spun twice.

Write a list of all possible combinations of outcomes and find the probability of the sum being less than 4.

P(\text { sum less than } 4)=\cfrac{3}{25}.

Teaching tips for compound probability

Review how to represent the probability of simple events with diagrams, before introducing the diagrams for the probability of compound events.

Before using worksheets, start with compound events that students can explore in the real world, like coin flips and rolling a die.

Provide easily accessible tutorials for students who are struggling or need additional support.

Students should have ample time to solve problems using sample spaces, before introducing the compound probability formula.

Easy mistakes to make

Leaving out or repeating outcomes when making a list
A common mistake is to accidentally leave out some outcomes or list the same one twice. This can mess up the total count. To avoid this, it helps to be organized when making the list. Using a table can make it easier to keep track and make sure you include each outcome only once.

Mixing up the order of the ratio when finding probability
When calculating probability, put the number of desired outcomes on top (numerator) and the total number of possible outcomes on the bottom (denominator). Getting this order wrong can lead to the wrong probability for an event.

Dependent events
When answering a question about compound events, always think carefully about whether the events are independent or dependent. For dependent events, the probability of the second event will change depending on the outcome of the first event.

It is a common error for students not to realize events are dependent and miscalculate the probability of the second event.

Practice compound probability questions

Compound probability FAQs

How can the probability of an event be represented?

It can be represented in fraction, decimal or percentage form.

What is simple probability?

It is the probability of a single event.

What are mutually inclusive events?

They are events that can happen at the same time. For example, when rolling a die, the outcomes of rolling an odd number or a 3 are mutually inclusive, because they can both happen.

What are mutually exclusive events?

They are events that cannot happen at the same time. For example, when rolling a die, the outcomes of rolling an odd number or an even number are mutually exclusive, because they cannot both happen.

See also: Mutually exclusive events

What is conditional probability?

The probability that event B will happen, given that event A has already happened. In other words, the chance of B happening, if A happens first.

See also: Conditional probability

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Compound Probability - Math Steps, Examples & Questions

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Free compound probability math topic guide, including step-by-step examples, free practice questions, teaching tips and more!

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Students complete a quick 3-question exit ticket: 1) Define probability in your own words, 2) Calculate P(rolling an even number on a six-sided die), 3) Explain one difference between theoretical and experimental probability.

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Throughout the lesson, observe student discussions during guided practice. Listen for correct use of probability vocabulary and logical reasoning when explaining their calculations and predictions.

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End-of-unit quiz includes: calculating simple probabilities, interpreting probability as fractions/decimals/percentages, comparing theoretical vs experimental results, and solving one real-world application problem.

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Mathematics Grade 10 Probability 😊📚🎲 Quiz

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Easily share the Quiz with students and get instant feedback. Great for practice, review, and classroom assessments.

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Students share in small groups: What was the most surprising discovery about probability today? How might you use probability concepts outside of math class? Groups can post their insights on a shared Padlet for the whole class to review.

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Each student writes a brief summary answering: What is probability? How do you calculate it? What's one example of probability in your daily life? This helps consolidate learning and identify any remaining misconceptions.

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Complete 10 probability problems involving dice, cards, and spinners. Include both calculation and interpretation questions. Problems progress from basic to intermediate difficulty levels.

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Students choose a real-world scenario (weather, sports, games) and research actual probability data. Write a one-paragraph reflection comparing what they expected vs. actual probabilities they found.

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Probability is used in all types of areas in real life including weather forecasting, sports betting, investing, and more.

The following examples share how probability is used in 10 real-life situations on a regular basis.

Example 1: Weather Forecasting

Perhaps the most common real life example of using probability is weather forecasting.

Probability is used by weather forecasters to assess how likely it is that there will be rain, snow, clouds, etc. on a given day in a certain area.

Forecasters will regularly say things like "there is an 80% chance of rain today between 2PM and 5PM" to indicate that there's a high likelihood of rain during certain hours.

Example 2: Sports Betting

Probability is heavily used by sports betting companies to determine the odds they should set for certain teams to win certain games.

For example, a sports betting company may look at the current record of two teams and determine that team A has a 90% probability of winning while team B has just a 10% probability of winning.

Based on these probabilities, the company would offer a higher payout for people who bet on team B to win since it's highly unlikely that team B will actually win.

Political forecasters use probability to predict the chances that certain candidates will win various elections.

For example, a forecaster might say that candidate A has a 60% chance of winning, candidate B has a 20% chance of winning, candidate C has a 10% chance of winning, etc. to give voters an idea of how likely it is that each candidate will win.

Note: A real-life example of a site that uses probability to perform political forecasting is FiveThirtyEight.

Example 4: Sales Forecasting

Many retail companies use probability to predict the chances that they'll sell a certain amount of goods in a given day, week, or month.

This allows the companies to predict how much inventory they'll need. For example, a company might use a forecasting model that tells them the probability of selling at least 100 products on a certain day is 90%.

This means they'll need to make sure they have at least 100 products on hand to sell (or preferably more) so they don't run out.

Example 5: Health Insurance

Health insurance companies often use probability to determine how likely it is that certain individuals will spend a certain amount on healthcare each year.

For example, a company might use factors like age, existing medical conditions, current health status, etc. to determine that there's a 90% probability that a certain individual will spend $10,000 or more on healthcare in a given year.

Individuals who are likely to spend more on healthcare will be charged higher premiums because the insurance company knows that they'll be more expensive to insure.

Example 6: Grocery Store Staffing

Grocery stores often use probability to determine how many workers they should schedule to work on a given day.

For example, a grocery store may use a model that tells them there is a 75% chance that they'll have more than 800 customers come into the store on a given day.

Based on this probability, they'll schedule a certain amount of workers to be at the store on that day to handle that many customers.

Example 7: Natural Disasters

The environmental departments of countries often use probability to determine how likely it is that a natural disaster like a hurricane, tornado, earthquake, etc. will strike the country in a given year.

If the probability is quite high, then the department will make decisions about housing, resource allocation, etc. that will minimize the effects done by the natural disaster.

Ordinary people use probability every day when they decide to drive somewhere.

Based on the time of day, location in the city, weather conditions, etc. we all tend to make probability predictions about how bad traffic will be during a certain time.

For example, if you think there's a 90% probability that traffic will be heavy from 4PM to 5:30PM in your area then you may decide to wait to drive somewhere during that time.

Investors use probability to assess how likely it is that a certain investment will pay off.

For example, a given investor might determine that there is a 1% chance that the stock of company A will increase 100x during the upcoming year.

Based on this probability, the investor will decide how much of their net worth to invest in the stock.

Probability is routinely used by anyone who plays card games on a regular basis.

For example, professional poker players use probability to determine how likely it is that a certain hand of cards will win and this informs them on how much they should bet.

If a player knows that there is a high probability that they will win a certain hand based on their cards, they will be more likely to bet more money.

Conversely, if they think the probability that they'll win is low then they may bet significantly less money.

The following tutorials provide additional information about probability:

Examples of Using Conditional Probability in Real Life
Probability vs. Proportion: What's the Difference?
Probability vs. Likelihood: What's the Difference?
Law of Total Probability: Definition & Examples

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10 Examples of Using Probability in Real Life - Statology

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Probability is used in all types of areas in real life including weather forecasting, sports betting, investing, and more. The following examples share how probability is used in 10 real-life situations on a regular basis. Perhaps the most common real life example of using probability is weather forecasting.

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Play an interactive probability game online for 15 minutes. Focus on games involving dice, coins, or cards that reinforce the concepts learned in class. Record your experimental results.

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In this game questions will appear in the form of "How likely is it that..." along with diagrams.

You can select your answer from four options: Certain, Probable, Unlikely and Impossible.

The first part of this game consists of some trials in which colored balls are shuffeled inside the machine.

In every trial, two colored balls are given, and you have to estimate the chance of them being picked by the machine.

The second part consists of a simulation: Balls are shuffeled inside the machine, and every time you pull the handle, a ball is picked. There is a counter for every color, and you get to see the relative frequency of the colors that were picked.

This quiz has 10 multiple-choice questions.

Click the correct answer for every question, and when you finish click the "Check Score" button.

Then you will get a full feedback where you can see your score and your correct and wrong answers, as well as a full detailed solution for every question.

This game consists of 6 different fun activities.

In these activities you can win tickets in carnival games, by making the correct calculations, and choosing the events with the highest chance of occuring.

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Interactive Probability Online Games for Kids

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In this game questions will appear in the form of "How likely is it that..." along with diagrams. You can select your answer from four options: Certain, Probable, Unlikely and Impossible. The first part of this game consists of some trials in which colored balls are shuffeled inside the machine.