Probability notation

Here you will learn about probability notation, including set notation.

Students will first learn about probability notation as part of statistics and probability in 7 th grade.

What is probability notation?

Probability notation is an efficient way of writing the probability of events happening or not happening.

To do this we use set notation, which is used when working with Venn diagrams. Events are usually notated using capital letters, as well as the use of some Greek letters.

Let’s look at some examples.

If you were to roll a fair six-sided dice and event A is rolling a six, a more efficient way of writing ‘what is the probability of rolling a 6 ’, you could write P(A) or P(6).

If you were to roll a fair six-sided dice and event A is rolling a six, a more efficient way of writing ‘what is the probability of not rolling a 6 ’, you could write P(A').

This links to Venn diagrams (shown below) as they show event A and anything that is not event A.

Step-by-step guide: Venn diagrams

Union

P(A\cup{B}) or ‘A union B’ is the chance of picking any outcome that is satisfied by event A or event B or both.

In a Venn diagram, it would be presented like this:

For example, ξ is the set of numbers 1-12. \, A and B are subsets of ξ where A= multiples of 3 and B= multiples of 4. One of the numbers is picked at random. Find the probability P(A\cup{B}).

Event A are the multiples of 3\text{:} \, 3, \, 6, \, 9, \, 12.

Event B are the multiples of 4\text{:} \, 4, \, 8, \, 12.

The numbers 1, \, 2, \, 5, \, 7, \, 10 and 11 are not in event A or event B.

There are 12 possible outcomes, the numbers 1 to 12.

There are a total of 6 possible outcomes (12 is only counted once even though it satisfies event A and event B).

So P(A\cup{B})=\cfrac{6}{12}.

Note: You can simplify probabilities if specified in the question but it can be easier to leave them in their original, unsimplified form if they are used in further calculations.

Intersection

P(A\cap{B}) or ‘A intersection B’ is the chance of picking any outcome that is satisfied by event A and event B, the joint probability of the events.

In a Venn diagram, it would be presented like this:

For example, ξ is the set of numbers 1-12. \, A and B are subsets of ξ where A= multiples of 3 and B= multiples of 4. One of the numbers is picked at random. Find the probability P(A\cap{B}).

Event A are the multiples of 3\text{:} \, 3, \, 6, \, 9, \, 12.

Event B are the multiples of 4\text{:} \, 4, \, 8, \, 12.

The numbers 1, \, 2, \, 5, \, 7, \, 10 and 11 are not in event A or event B.

There are 12 possible outcomes, the numbers 1 to 12.

There is only 1 number that satisfies event A and event B, the number 12.

So P(A\cap{B})=\cfrac{1}{12}.

Mutually exclusive and non-mutually exclusive events

Mutually exclusive events cannot happen at the same time, for example, rolling a 4 and a 5 on a standard 6 -sided dice.

Non-mutually exclusive events can happen at the same time, for example, rolling a 4 and an even number on a standard 6 -sided dice. It is necessary to determine whether events are therefore mutually exclusive when considering the union of events.

If events are mutually exclusive, then P(A\cup{B})=P(A)+P(B).

If events are not mutually exclusive, then P(A\cup{B})=P(A)+P(B)-P(A\cap{B}).

Step-by-step guide: Mutually Exclusive Events

Here are some more common probability notations, ranging from basic notations in middle school to more complex notations in high school and beyond:

What is probability notation?

Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – Statistics and Probability (7.SP.C.5)
  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood.
  
  A probability near 0 indicates an unlikely event, a probability around \cfrac{1}{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

How to use probability notation

In order to use probability notation to calculate the probability of an event:

1. Identify the event(s) needed.
2. Identify how many possible outcomes there are.
3. Identify how many times the event(s) occur.
4. Write this as a probability.

Probability notation examples

Example 1: probability of an event

A box contains a variety of 2D shapes:

Event A\text{:} selecting a circle.

Event B\text{:} selecting a square.

Event C\text{:} selecting a triangle.

Event D\text{:} selecting a star.

Calculate P(C).

1. Identify the event(s) needed.

Event C is selecting a triangle.

2Identify how many possible outcomes there are.

There are 12 possible outcomes as there are 12 shapes in the box.

3Identify how many times the event(s) occur.

There are 3 triangles so the expected value is 3.

4Write this as a probability.

Example 2: probability of an event not happening

A bowl of fruit contains: 3 apples, 2 oranges, 3 bananas, and 2 plums.

Event A\text{:} picking an apple.

Event B\text{:} picking a banana.

Event C\text{:} picking a plum.

Event D\text{:} picking an orange.

What is P(D')?

Example 3: probability of event A or event B or both

A fast food restaurant records what their customers ate on Monday between 10 am and 11 am. This data is used to predict what customers will buy at the same time on the following Monday.

Calculate P(A\cup{C}) .

Example 4: probability of event A or event B or both

A fair, eight sided spinner is spun.

Event A\text{:} land on 3.

Event B\text{:} land on an even number.

Calculate P(A\cup{B}).

Example 5: probability of event A and event B

A fair, eight sided spinner is spun.

Event A\text{:} land on an odd number.

Event B\text{:} land on a prime number.

Calculate P(A\cap{B}).

Example 6: probability of event A and event B

A standard, fair, 6 -sided dice is rolled.

Event A\text{:} Roll a multiple of 3.

Event B\text{:} Roll a 1.

Calculate P(A\cap{B}).

Teaching tips for probability notation

• Introduce and explain the statistical symbols used in probability notation, such as P(A), \, P(B|A), \, A\cup{B}, \, A\cap{B}, and A'.

• Compare and contrast different notations used in probability theory, such as conditional probability versus joint probability.

• Provide exercises that follow a linear progression, where each problem builds on the understanding gained from the previous one. This helps reinforce learning and shows the interconnectedness of probability concepts.

• Use simple examples to introduce the concept of the empty set (∅). For example, explain that if you are rolling a fair die, the event of rolling a 7 is an empty set because it is impossible.

Easy mistakes to make

• Confusing ‘union’ and ‘intersection’
  P(A\cup{B}) and P(A\cap{B}) are similar and can easily be interchanged due to confusion.

• Assuming all expected values are \textbf{A}\cup\textbf{B}
  Some values may be in the universal set but not be satisfied by event A or event B, it might sit outside.
  
  For example, this Venn diagram displays two numbers, 1 and 9, that are not satisfied by event A or event B but are in the universal set. The universal set is notated with the Greek letter ξ.
  
  

• Incorrect use of and/or rule for independent events and conditional probability
  When calculating the probability of event A or event B, you need to add the probabilities.
  
  When calculating the probability of event A and event B, you need to multiply the probabilities.

• Misinterpreting the notation
  Set notation includes the use of capital letters, lowercase letters, and Greek letters. These are specific and cannot be interchanged.
  
  For example, set ‘A’ could be even numbers, and set ‘a’ could be odd numbers.

• Not listing outcomes
  When there are two or more events we need to use systematic listing strategies to include all the permutations.

Related probability lessons

• Probability formula
• Independent events
• Mutually exclusive events
• Probability scale
• Tree diagram probability

The next lessons are

• Compound events
• Probability distribution
• Units of measurement

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs.