Instead of writing a word or phrase to talk about the probability of an event like 'flipping a head' or P\text{(head)}, we often represent an event with a single letter, for example, representing the event of flipping a head as H. This can make it is easier to talk about this event. We can then talk about the probability of that event as being P(H), rather than writing P\text{(head)} every time.

A Venn diagram with 2 circles for sets A and B that overlap. The area inside circle A is shaded.

To investigate some further notation let's consider two events: A and B. A is the event of you seeing an asteroid tonight. B is the event that you will have a burger tonight. This Venn diagram is of the two events with the area of P(A) highlighted.

A Venn diagram with 2 circles for sets A and B that overlap. The area outside circle A is shaded.

The complement of the probability of A is given by 1-P(A). We also have annotation for the complement of the event itself, A', it may also be written as \overline{A}. Given our event, A, the event A' would be the event not seeing an asteroid tonight.

This Venn diagram has the area of P(A') highlighted.

A Venn diagram with 2 circles for sets A and B that overlap. The area inside the 2 circles is shaded.

The notation \cup used as P(A \cup B) means the probability of A or B happening. This area is shown in the Venn diagram.

It is the probability of:

You seeing an asteroid but not eating a burger
You eating a burger but not seeing an asteroid
You seeing an asteroid and eating a burger

A Venn diagram with 2 circles for sets A and B that overlap. The intersection of the circles is shaded.

The notation \cap used as P(A \cap B) means the probability of A and B both happening. This area is shown in the Venn diagram.

It is the probability of:

You seeing an asteroid and eating a burger

Examples

Example 1

Two events are defined as:

Event A: it will rain tomorrow
Event B: there will be a storm tomorrow

The notation P(A \text{ or } B) is suitable to describe which of the following probabilities?

Probability of no rain tomorrow

Probability of there being either a storm or rain tomorrow

Probability of a storm occurring tomorrow

Probability of there being a storm but no rain

Worked Solution

Create a strategy

Choose the option that describes both events will occur.

Apply the idea

Options A and C are incorrect since they only mentioned one event.

Option D is incorrect because a storm is the only one that will occur.

The correct option is B: Probability of there being either a storm or rain tomorrow.

Example 2

In an experiment, a number is chosen randomly from the numbers listed below: \{2,\,3,\,5,\,6,\,7,\,10,\,12,\,14,\,15,\,16,\,19,\,20\}

Event A = an odd number is chosen
Event B = a multiple of 4 is chosen

Which of the following has the largest probability?

A \cup B

A \cap B

Worked Solution

Create a strategy

Compare the number of the outcome of each notation and choose the one with the greatest number of outcomes.

Apply the idea

\displaystyle B'	\displaystyle =	\displaystyle \{2,\,3,\,5,\,6,\,7,\,10,\,14,\,15,\,19\}	Numbers that are not a multiple of 4
\displaystyle A \cup B	\displaystyle =	\displaystyle \{3,\,5,\,7,\,12,\,15,\,16,\,19,\,20\}	Numbers that are a multiple of 4 or odd
\displaystyle B	\displaystyle =	\displaystyle \{12,\,16,\,20\}	Numbers that are multiples of 4
\displaystyle A \cap B	\displaystyle =	\displaystyle \{\}	No numbers are odd and multples of 4
\displaystyle A	\displaystyle =	\displaystyle \{3,\,5,\,7,\,15,\,19\}	Odd numbers

B' has the greatest number of outcomes, so the correct answer is option A: B'.

Which of the following has a value of 0?

P(A \cap B)

P(A' \cup B)

P(A \cap B')

P(A' \cap B')

Worked Solution

Create a strategy

Find the probability of each option. Use the formula: P(E)=\dfrac{\text{Number of favourable events}}{\text{Total number of events}}

Apply the idea

There were 12 numbers in the original list, so this will be the denominator in our probabilities.

\displaystyle P(A \cap B)	\displaystyle =	\displaystyle \dfrac{0}{12}	No numbers satisfy A \cap B
	\displaystyle =	\displaystyle 0	Evaluate
\displaystyle P(A' \cup B)	\displaystyle =	\displaystyle \dfrac{7}{12}	7 numbers are not odd or multiples of 4
\displaystyle P(A \cap B')	\displaystyle =	\displaystyle \dfrac{5}{12}	5 numbers are odd and not multiples of 4
\displaystyle P(A' \cap B')	\displaystyle =	\displaystyle \dfrac{4}{12}	4 numbers are not odd and not multiples of 4

The correct answer is option A: P(A \cap B).

Example 3

A student creates the following diagram of their favourite animals.

The event F is: "selecting a favourite four legged animal".

The event S is : "selecting a favourite animal with stripes".

A Venn Diagram with 2 circles for sets Four legs and stripes that overlap. Ask your teacher for more information.

Draw a Venn diagram and shade the region that represents the favourable outcomes for the probability P(F).

Worked Solution

Create a strategy

Draw two sets with titles F and S, and then shade the circle for F.

Apply the idea

A Venn Diagram with 2 circles for sets F and S that overlap. The area inside circle F is shaded.

Draw a Venn diagram and shade the region that represents the favourable outcomes for the probability P(S').

Worked Solution

Create a strategy

Draw two sets with titles F and S, and then shade the region outside the S circle.

Apply the idea

A Venn Diagram with 2 circles for sets F and S that overlap. Set F and the area outside the sets is shaded.

Idea summary

Venn diagrams can be a useful way to think about these notations.

A - a single letter can represent an event.
A' - the complement of event A.
P(A) - the probability of event A happening.
P(A \cup B) - the probability that either of A or B happening.
P(A \cap B) - the probability that both A and B happening.

8.01 Compound events

Ideas

Probability notation

Examples

Example 1

Example 2

Example 3

Outcomes

MA5.2-17SP

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