# 9.3a Finding the Probability of a Combined Event by Listing the Outcomes

9.3 Probability of a Combined Event

9.3a Finding the Probability of a Combined Event by Listing the Outcomes
1. A combined event is an event resulting from the union or intersection of two or more events.
2. The union of combined event ‘A or B’ = A υ B
3. The intersection of combined event ‘A and B’ = B

Example:
Diagram below shows five cards labelled with letters.

All these cards are put into a box. A two-letter code is to be formed by using any two of these cards. Two cards are picked at random, one after another, without replacement.
(a) List all sample space
(b) List all the outcomes of the events and find the probability that
(i) The code begins with the letter P.
(ii) The code consists of two vowel or two consonants.

Solution:
(a)
Sample space, S
= {(G, R), (G, A), (G, P), (G, E), (R, G), (R, A), (R, P), (R, E), (A, G), (A, R),
(A, P), (A, E), (P, G), (P, R), (P, A), (P, E), (E, G), (E, R), (E, A), (E, P)}

(b)
n(S) = 20
Let
A = Event of choosing a code begins with the letter P
B = Event of choosing the code consists of two vowel or two consonants.

(i)
A = {(P, G), (P, R), (P, A), (P, E)}
n(A) = 4
$P\left(A\right)=\frac{4}{20}=\frac{1}{5}$

(ii)
B = {(G, R), (G, P), (R, G), (R, P), (A, E), (P, G), (P, R), (E, A)}
n(B) = 8
$P\left(B\right)=\frac{8}{20}=\frac{2}{5}$