Question

Assume that 2 cards are drawn in succession and without replacement from a standard deck of 52 cards. Find the probability that the following occurs. (Enter your probability as a fraction.) The second card is a 2, given that the first card was a 2.

Answer 1

Solution

Let A represent the event that the first card is a 2 and B represent the event that the second card is a 2.

Back-up Theory

Probability of an event E, denoted by P(E) = n/N ………                                ……………......................................…….…(1)

where n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and

N = n(S) = Total number all possible outcomes/cases/possibilities.

If A and B are two events such that probability of B is influenced by occurrence or otherwise of A, then Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)………….............................................….. .….(2)

Now to work out the solution,

Probability the second card is a 2, given that the first card was a 2

= P(B/A)

= P(B ∩ A)/P(A) [vide (2)] …………………………………………………............................................……………………. (3)

Now, B ∩ A => both the first and the second cards are 2’s. There are four 2’s, namely 2 of clubs, 2 of diamonds, 2 of spades and 2 of hearts. Of these 4, any one can come as the first card. Since the drawing is without replacement, the second card can only be any one of the remaining three cards of 2’s. Thus,

B ∩ A is possible in (4 x 3) = 12 ways => vide (1), n = 12 ……………...........................................……………………….. (4)

Two cards can be drawn without replacement in (52 x 51) ways.=> vide (1), N = (52 x 51)…......................................... (5)

(1), (4) and (5) => P(B ∩ A) = 12/(52 x 51) = 1/(13 x 17) ………………………..........................................………………. (6)

A => first cars is a 2. Out of 52 cards, there are four cards that are 2’s. So, vide (1),

P(A) = 4/52 = 1/13 ……………………………………………………..............................................…………………………. (7)

(6) and (7) in (3) gives: P(B/A) = {1/(13 x 17)}/( 1/13)

= 1/17 Answer

DONE