9.3b Finding the Probability of Combined Events (a) A or B (b) A and B

9.3 Probability of a Combined Event

9.3b Finding the Probability of Combined Events (a) A or B (b) A and B

1. The probability of a combined event ‘A or B’ is given by the formula below.

\begin{array}{l} P (A or B) = P (A \cup B) \\ = \frac{n (A \cup B)}{n (S)} \end{array}

2. The probability of a combined event ‘A and B’ is given by the formula below.

\begin{array}{l} P (A and B) = P (A \cap B) \\ = \frac{n (A \cap B)}{n (S)} \end{array}

Example:

The probabilities that two Form 5 students, Fiona and Wendy will pass the English oral test are

\frac{1}{3} and \frac{2}{5}

respectively. Calculate the probability that

(a) both Fiona and Wendy past the English oral test,
(b) both Fiona and Wendy fail the English oral test,

(c) either one of them passes the English oral test,
(d) at least one of them passes the English oral test.

Solution:

Let

F = Event that Fiona passes the English oral test

W = Event that Wendy passes the English oral test

Therefore,

F’ = Event that Fiona fails the English oral test

W’ = Event that Wendy fails the English oral test

\begin{array}{l} P (F) = \frac{1}{3}, P (F ‘) = \frac{2}{3} \\ P (W) = \frac{2}{5}, P (W ‘) = \frac{3}{5} \end{array}

(a)

P (both Fiona and Wendy past the English oral test)

= P (F ∩ W)

= P (F) x P (W)

\begin{array}{l} = \frac{1}{3} \times \frac{2}{5} \\ = \frac{2}{15} \end{array}

(b)

P (both Fiona and Wendy fail the English oral test)

= P (F’ ∩ W’)
= P (F’) x P (W’)

\begin{array}{l} = \frac{2}{3} \times \frac{3}{5} \\ = \frac{2}{5} \end{array}

(c)

P (either one of them passes the English oral test)

= P (F ∩ W’) + P (F’ ∩ W)
= (P (F) x P (W’)) + (P (F’) x P (W))

\begin{array}{l} = (\frac{1}{3} \times \frac{3}{5}) + (\frac{2}{3} \times \frac{2}{5}) \\ = \frac{7}{15} \end{array}

(d)

P (at least one of them passes the English oral test)

= 1 – P (Both of them fail) ← (concept of complement event)

= 1 – P (F’) x P (W’)

\begin{array}{l} = 1 - \frac{2}{5} \\ = \frac{3}{5} \end{array}