7.5 SPM Practice (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)}
$\text{Probability}=\frac{2}{12}=\frac{1}{6}$

Question 8 (6 marks):
A bag contains five cards, labelled with the letters I, J, K, M and U.
One card is picked at random from the bag and the letter is recorded. Without replacement, another card is picked at random from the bag and the letter is also recorded.

(a) Complete the sample space in the answer space.

(b) By listing all the possible outcomes of the event, find the probability that

(i) the first card picked is labelled with a vowel.

(ii) the first card picked is labelled with a consonant and the second card picked is labelled with a vowel.

Answer:
{(I, J), (I, K), (I, M), (I, U), (J, I), (J, K), (J, M), (J, U), (K, I), (K, J), (   , ), (   , ), ( , ), ( , ), (   , ), (   ,   ), (   , ), (   ,   ), (   , ), (   ,   )}


Solution:
(a)
S = {(I, J), (I, K), (I, M), (I, U), (J, I), (J, K), (J, M), (J, U), (K, I), (K, J), (K, M), (K, U), (M, I), (M, J), (M, K), (M, U), (U, I), (U, J), (U, K), (U, M)}

(b)(i)
{(I, J), (I, K), (I, M), (I, U), (U, I), (U, J), (U, K), (U, M)}
$\begin{array}{l}\text{Probability of first card with vowels}\\ =\frac{8}{20}\\ =\frac{2}{5}\end{array}$

(b)(ii)
{(J, I), (J, U), (K, I), (K, U), (M, I), (M, U)}
$\begin{array}{l}\text{Probability}\\ =\frac{6}{20}\\ =\frac{3}{10}\end{array}$