9.5.4 Probability of Combined Events, SPM Paper (Long Questions)

Question 7:
Diagram below shows 4 cards labelled with letters.


Two cards are chosen at random from the box, one by one, without replacement.
(a) List the sample space.
(b) Find the probability that
(i) at least one card chosen is labelled C,
(ii) both cards chosen are labelled with same letter.


Solution:
(a)
Sample space, S
= {(C, O₁), (C, O₂), (C, L), (O₁, C), (O₁, O₂), (O₁, L), (O₂, C), (O₂, O₁), (O₂, L), (L, C), (L, O₁), (L, O₂)}
n(S) = 12

(b)(i)
At least one card chosen is labelled C
= {(C, O₁), (C, O₂), (C, L), (O₁, C), (O₂, C), (L, C)}
$\text{Probability}=\frac{6}{12}=\frac{1}{2}$

(b)(ii)
Both cards chosen are labelled with same letter
= {(O₁, O₂), (O₂, O₁)}
$\text{Probability}=\frac{2}{12}=\frac{1}{6}$

$=\frac{7}{9}$