Compound probability

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.

What is compound probability?

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?

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

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

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

There are 5 types of sandwiches and 8 different desserts.

3Multiply 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

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?

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?

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.

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.

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

Start by listing all the art classes. Then from each art class, create 3 branches for the foreign language classes.

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

There is one possible outcome that is chorus and French.

The probability is the ratio of the desired subset (chorus and French) over the total possible outcomes.

\mathrm{P}(\text { chorus and French })=\cfrac{1}{12}.

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.

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

Each spin has 5 outcomes, all equally likely. You can use a table to show all combinations of the spinners.

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

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

There are three possible outcomes with a sum less than 4.

The probability is the ratio of the desired subset (sum less than 4 ) over the total possible outcomes.

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.

The next lessons are

• Probability distribution
• Units of measurement
• Represent and interpret data

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