POP

E8.1 Introduction to Probability: Notation & the Complement Rule

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Probability measures how likely an event is, on a scale from $0$ (impossible) to $1$ (certain).

01 — Explanation

What does P(A) mean, and what values can it take?

$P(A)$ is the probability that event $A$ occurs. It can be given as a fraction, decimal, or percentage, but it must always satisfy $0 \le P(A) \le 1$ — any answer outside this range signals an arithmetic error somewhere in the working.

02 — Explanation

How do I find the probability of a single equally likely outcome?

When every outcome in a situation is equally likely, the probability of an event is the number of ways it can happen divided by the total number of possible outcomes:

P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}

A bag contains $4$ red, $5$ blue, and $3$ green counters. Choosing one counter at random, $P(\text{blue}) = \dfrac{5}{12}$ , since there are $12$ counters in total.

03 — Explanation

What is the complement of an event, and how do I use it?

The complement of $A$ , written $A'$ , is the event that $A$ does not happen. Since something either happens or doesn't, the two probabilities must add to $1$ :

P(A') = 1-P(A)

P(B)=0.62 \implies P(B')=1-0.62=0.38

Exam Tip

The prime symbol $'$ always means "NOT this event" — reading $P(B')$ as simply "the probability of $B$ " is a common misreading that throws away an easy mark.

04 — Explanation

How do I combine the complement rule with information from a Venn diagram?

Treat "probability × total" as giving you an expected count for any region of the diagram, and use this to check that different pieces of given information are consistent with each other.

In a group of $50$ people, $P(B\cap C)=0.36$ where $B$ is "owns a bicycle" and $C$ is "owns a car." Multiplying by the group size:

n(B\cap C) = 0.36\times50=18

If the question separately states that $18$ people own both, this confirms the two pieces of information agree.

05 — Explanation

How do I use probability notation with a two-way table?

Read the row and column totals directly as counts, then convert to a probability by dividing by the grand total, exactly as you would with a simple fraction. Two-way tables are especially useful for questions phrased as "given that," since restricting to one row or one column narrows the total you divide by.

Exam Tip

Whenever a probability question includes the phrase "given that," you are being asked for a conditional probability — restrict your total to only the row or column that satisfies the given condition, rather than dividing by the grand total of the whole table.

Worked Example

In a class of $30$ students, $P(\text{studies French}) = 0.4$ .

(a) Find the probability that a randomly chosen student does not study French.

(b) Find the number of students who do not study French.

(a) Apply the complement rule: $P(\text{not French}) = 1-0.4 = 0.6$ .
(b) Multiply this probability by the total number of students: $0.6\times30=18$ students.

Exam Tip

Whenever a question gives a probability and a total group size together, check whether it's actually asking for a probability or a count — multiplying a probability by the total converts one into the other, and dividing a count by the total converts it back.