# 7.3 Probability of an Event

(A) Probability of an Event
1. The probability of an event A, P(A) is given by

2. If P(A) = 0, then the event A will certainly not occur.
3. If P(A) = 1, then the event A will certainly to occur.

Example 1:
Table below shows the distribution of a group of 80 pupils playing a game.

 Form Four Form Five Girls 28 16 Boys 12 24
A pupil is chosen at random from the group to start the game.
What is the probability that a boy from Form Five will be chosen?

Solution:
Let
= Event that a boy from Form Five
= Sample space
n(S) = 28 + 12 + 16+ 24 = 80
n(A) = 24
$\begin{array}{l}P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}\\ \text{}=\frac{24}{80}=\frac{3}{10}\end{array}$

(B) Expected Number of Times an Event will Occur
If the probability of an event A and the number of trials are given, then the number of times event occurs
= P(A) × Number of trials

Example 2:
In a football training session, the probability that Ahmad scores a goal in a trial is ⅝. In 40 trials are chosen randomly, how many times is Ahmad expected to score a goal?

Solution:
Number of times Ahmad is expected to score a goal
= ⅝ × 40
= 25

(C) Solving Problems
Example 3:
Kelvin has 30 white, blue and red handkerchiefs. If a handkerchief is picked at random, the probability of picking a white handkerchief is $\frac{2}{5}.$  Calculate
(a) the number of white handkerchiefs.
(b) the probability of picking a blue handkerchief if 8 of the handkerchiefs are red in colour.

Solution:
Let
= Event that a white handkerchief is picked.
= Event that a blue handkerchief is picked.
= Event that a red handkerchief is picked.
= Sample space

(a)

n(S) = 30
$\begin{array}{l}n\left(W\right)=P\left(W\right)×n\left(S\right)\\ \text{}=\frac{2}{5}×30=12\end{array}$

(b)

Given n(R) = 8
n(B) = 30 – 12 – 8 = 10
$\begin{array}{l}P\left(B\right)=\frac{n\left(B\right)}{n\left(S\right)}\\ \text{}=\frac{10}{30}=\frac{1}{3}\end{array}$