Question

1. Two cards are drawn from a deck of cards (consisting of 52 cards: 13 cards for each of the four suits, Spades (S), Hearts (H), Diamonds (D) and Clubs (C)). Drawing is done with replacement, that is, the first card is drawn, recorded and put back in the deck, then the second card is drawn and recorded. Assuming an outcome is recorded in the order which two cards are drawn, what is the sample space for this (random) phenomenon? Write all possible outcomes.

Assuming each card is likely to be drawn, and two draws are independent, what is the probability that two cards are of the same suit?(**) Write the last calculation formula and the final answer value.

Answer 1

Sample space S = {(S, S), (S, H), (S, D), (S, C), (H, S), (H, H), (H, D), (H, C), (D, S), (D, H), (D, D), (D, C), (C, S), (C, H), (C, D), (C, C)}

Coming to the second part of question.

Let's find the number of ways 2 cards can be drawn with replacement.

There are 52 cards in the deck. Hence, the first card can be drawn in 52 ways. Now this card is replaced in the deck so the number of cards in the deck is again 52. Hence, the second card can also be darwn in 52 ways. So 2 cards (one-by-one and with replacement) can be drawn in 52*52 ways.

Now, let's find the number of ways in which 2 Spades can be drawn one-by-one and with replacement. There are 13 Spades in the deck. Hence, there are 13 ways in which a Spade can be drawn. After drawing the Spade, it is replaced in the deck. So there are again 13 Spades in the deck. Hence, there are again 13 ways to draw the second Spade from the deck. So 2 Spades (one-by-one and with replacement) can be drawn in 13*13 ways.

Similarly, 2 Hearts (one-by-one and with replacement) can be drawn in 13*13 ways.

2 Diamonds (one-by-one and with replacement) can be drawn in 13*13 ways.

2 Clubs (one-by-one and with replacement) can be drawn in 13*13 ways.

Hence, Probability that the 2 cards drawn with replacement of the same suit = (Number of ways 2 cards of same colour are drawn with replacement) / (Total number of ways any 2 cards can be drawn with replacement)

= (13*13 + 13*13 + 13*13 + 13*13)/(52*52)

= (4*13*13) / (52*52) = 1/4 = 0.25