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Probability scale

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Expected value

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

There are also probability notation* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**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. Below are two Venn diagrams which involve two sets and show event A and anything that is not event A.

**Step-by-step guide:** Venn diagrams

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, \xi is the set of numbers, 1-12. \ A and B are subsets of \xi .

A = multiples of 3.

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 = 3, 6, 9, 12 .

Event B are the multiples of 4 = 4, 8, 12 .

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

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

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

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

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, \xi is the set of numbers, 1-12. \ A and B are subsets of \xi .

A = multiples of 3.

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 = 3, 6, 9, 12 .

Event B are the multiples of 4 = 4, 8, 12 .

The numbers 1, 2, 5, 7, 10 and 11 are not in event A or 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)=\frac{1}{12}.

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

**Identify the event.****Identify how many possible outcomes there are.****Identify how many times the event occurs.****Write this as a probability.**

Get your free probability notation worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free probability notation worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONEvent A is picking a circle.

Event B is picking a square.

Event C is picking a triangle.

Event D is picking a star.

What is P(C)?

**Identify the event.**

Event C is the probability of picking a triangle.

2**Identify how many possible outcomes there are.**

There are 10 possible outcomes as there are 10 shapes in the bag.

3**Identify how many times the event occurs.**

There are two triangles so the expected value is 2.

4**Write this as a probability.**

Event A is picking a circle.

Event B is picking a square.

Event C is picking a triangle.

Event D is picking a star.

What is P(D’)?

Identify the event.

Event D is the probability of picking a star. In this example we want the complement of D so we want any shape that is not a star.

**Identify how many possible outcomes there are.**

There are 10 possible outcomes as there are 10 shapes in the bag.

**Identify how many times the event occurs.**

There are 9 shapes that are not a star.

**Write this as a probability.**

\frac{9}{10}

Event A is picking a blue shape.

Event B is picking a triangle.

Calculate P(A \cup B).

**Identify the outcomes that are event ** \bf{A} ** or event ** \bf{B} **.**

A blue shape or a triangle needs to be picked.

**Identify how many possible outcomes there are.**

There are 10 shapes in the bag.

**Identify how many times the event occurs.**

Event A: \ 3 shapes

Event B: \ 2 triangles

There are a total of 5 possible outcomes.

**Write this as a probability.**

P(A \cup B) = \frac{5}{10} \ or \ 0.5.

Event A the spinner landing on blue.

Event B is the spinner landing on an even number.

Calculate P(A \cup B).

**Identify the outcomes that are event ** \bf{A, B} **or both.**

The spinner needs to land on a blue section or a section numbered 2, 4, 6 or 8.

**Identify how many possible outcomes there are.**

There are 8 sections on the spinner.

**Identify how many times the event occurs.**

Event A: \ 3 sections

Event B: \ 2, 4, 6, 8

There are a total of 5 possible outcomes ( 4 and 6 satisfy event A and event B ).

**Write this as a probability.**

\frac{5}{8}

Event A is the spinner landing on blue.

Event B is the spinner landing on an even number.

Calculate P(A \cap B).

**Identify the outcomes that are event ** \bf{A} ** and event ** \bf{B} **.**

The spinner needs to land on a blue section and an even number.

**Identify how many possible outcomes there are.**

There are 8 sections on the spinner.

**Identify how many times the event occurs.**

Two sections are blue and an even number.

There are 2 possible outcomes.

**Write this as a probability.**

\frac{2}{8}

Event A is the spinner landing on red.

Event B is the spinner landing on an even number.

Calculate P(A \cap B).

**Identify the outcomes that are event ** \bf{A} ** and event ** \bf{B} **.**

The spinner needs to land on a red section and an even number.

**Identify how many possible outcomes there are.**

There are 8 sections on the spinner.

**Identify how many times the event occurs.**

The only red section is an odd number.

There are 0 possible outcomes.

**Write this as a probability.**

0

**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**\bf{A \cup 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 \xi.

**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 use of capital letters, lower case letters and greek letters. These are specific and cannot be interchanged.

For example, set ‘A’ could be the even numbers and set ‘a’ could be odd numbers.

**Listing outcomes**

When there are two or more events we need to use systematic listing strategies to include all the permutations.

1.

Event A is picking a blue marble.

Event B is picking a green marble.

Event C is picking a red marble.

Work out P(B).

\frac{5}{8}

\frac{3}{8}

\frac{3}{5}

\frac{1}{3}

The notation P(B) means we want the probability of event B. Event B is picking a green marble.

There are 5 green marbles and 8 marbles in total.

So, P(B) = \frac{5}{8}.

2.

Event A is picking a blue marble.

Event B is picking a green marble.

Event C is picking a red marble.

Work out P(C’).

\frac{2}{8}

\frac{1}{4}

\frac{2}{3}

\frac{6}{8}

The notation P(C’) means we want the probability that it is not event C. Event C is picking a red marble.

There are two red marbles and six marbles that are not red.

There are 8 marbles in total.

So, P(C’) = \frac{6}{8}.

3. A fair six sided dice is rolled.

Event A is the dice landing on an even number.

Event B is the dice landing on ‘1’.

Calculate P(A \cup B).

\frac{1}{6}

\frac{3}{6}

\frac{4}{6}

1

The notation P(A \cup B) means in A, or B, or both. There are three even numbers and one ‘1’.

Therefore there are 4 possible outcomes that satisfy event A, event B, or both.

There are 6 outcomes in total.

So, P(A \cup B) = \frac{4}{6}.

4. A fair six sided dice is rolled.

Event A is the dice landing on an even number.

Event B is the dice landing on a prime number.

Calculate P(A \cup B).

\frac{1}{6}

\frac{5}{6}

\frac{3}{6}

0

The notation P(A \cup B) means in A, or B, or both. There are three even numbers (2, 4, 6) and three prime numbers (2, 3, 5).

‘2’ satisfies both event A and event B so there are 5 possible outcomes that satisfy event A and event B.

There are 6 outcomes in total.

So, P(A \cup B) = \frac{5}{6}.

5. A fair six sided dice is rolled.

Event A is the dice landing on an even number.

Event B is the dice landing on ‘2’.

Calculate P(A \cap B).

\frac{1}{6}

\frac{3}{6}

\frac{2}{6}

1

The notation P(A \cap B) means in A and B. There are three even numbers and one ‘2’.

Therefore there is 1 possible outcome that satisfies event A and event B.

There are 6 outcomes in total.

So, P(A \cap B) = \frac{1}{6}.

6. A fair six sided dice is rolled.

Event A is the dice landing on an odd number.

Event B is the dice landing on a prime number.

Calculate P(A \cap B).

\frac{1}{6}

\frac{3}{6}

\frac{2}{6}

0

The notation P(A \cap B) means in A and B. There are three odd numbers (1, 3, 5) and three prime numbers (2, 3, 5).

‘3 and 5′ satisfies both event A and event B so there are 2 possible outcomes.

There are 6 outcomes in total.

So, P(A \cap B) = \frac{2}{6}.

1. \xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}

A = square numbers

B = odd numbers

(a) Calculate P(B).

b) Calculate P(A \cup B).

**(2 marks)**

Show answer

(a)

P(B) = \frac{8}{15}

**(1)**

(b)

P(A \cup B) = \frac{9}{15}, or equivalent. For example, \frac{3}{5} or 0.6.

**(1)**

2. (a) Here is a Venn diagram.

(a) Calculate P(A’).

b) Calculate P(A \cap B).

**(2 marks)**

Show answer

(a)

P(A’) = 0.5 or equivalent. For example, \frac{5}{10} or \frac{1}{2}.

**(1)**

(b)

P(A \cap B) = 0.1 or equivalent. For example, \frac{1}{10}.

**(1)**

3. \xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

A = \{multiples of 4\}

\begin{aligned} &A \cap B = \{4, 12\} \\ &A \cup B = \{1, 2, 3, 4, 6, 8, 9, 12\} \end{aligned}

(a) Calculate P(B).

(b) Calculate P(A’).

**(2 marks)**

Show answer

(a)

P(B) = \frac{6}{12}, or equivalent. For example, \frac{1}{2} or 0.5.

**(1)**

(b)

P(A’) = \frac{9}{12}, or equivalent. For example, \frac{3}{4} or 0.75.

**(1)**

You have now learned how to:

- Interpret probability notation to calculate the probability of an event
- Apply probability notation to sets and Venn diagrams
- Interpret notation which includes capital letters, lower case letters and greek letters

- Probability distribution
- How to calculate probability
- Combined events probability

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