Question

Three cards are drawn successively from a pack of 52 playing cards and their colours are noted:
(i) Write down the sample space for this experiment.
(ii) Write down the subsets, of S corresponding to the events described below:
       (a) A: exactly two red cards are obtained.
       (b) B: at least two red cards are obtained.
       (c) C: at most two red cards are obtained.
       (d) D: at most one red card is obtained.

Answer 1

Hint: Here, we have two sub-questions, in the first sub-question we need to find the total number of the outcomes for the experiment conducted, we only need to consider two colours which are red and black and find the sample space accordingly. In the second sub-question, the subsets will be obtained with the help of the corresponding sample space.

Complete step-by-step answer:
(i) For the first question, we need to find the total number of outcomes which is also known as the sample space when three cards are drawn in sequence from 52 playing cards. There are 26 red cards and 26 black cards, so there are only two colour cards in the deck. Therefore, the sample space is,
S = {(R, R, R), (B, R, R), (B, B, R), (B, R, B), (R, B, B), (R, R, B), (R, B, R), (B, B, B)}
Therefore, The total number of outcomes, n(S) = 8
(ii) In the second part of the question, we have to find the subsets according to the question mentioned.
(a) Exactly two red cards are obtained.
Here, we need to check the sample space and its outcomes and write the events where there are only two red cards when the three cards are drawn.
A = {(B, R, R), (R, R, B), (R, B, R)}
n(A) = 3
(b) At least two red cards are obtained.
In this, we need to select the event where there are at least two red cards when drawn, which means, we will obtain two or more than two red cards.
B = {(B, R, R), (R, R, B), (R, B, R), (R, R, R)}
n(B) = 4
(c) At most two red cards are obtained.
In this, we need to get at most two red cards when three cards are drawn, which means we need to get not more than 2 red cards.
C = {(B, R, R), (R, R, B), (R, B, R), (B, B, B), (B, B, R), (B, R, B), (R, B, B)}
n(C) = 7
(d) At most 1 red card is obtained.
In this, we need to get at most one red card when three cards are drawn, which means we need to get not more than 1 red card.
D = {(B, B, B), (B, B, R), (B, R, B), (R, B, B)}
n(D) = 4
Hence, the total number of sample spaces is 8 and the total number of events in each subset, like n(A), n(B), n(C) and n(D) is 3, 4, 7 and 4 respectively.

Note: In the second sub-question, in the at most question, (B, B, B) does not have any red cards but it is still considered because at most means not more than, which indicates that (B, B, B) has 0 red cards which is not more than 2 red cards or not more than one red card.