probability

Question

the following exercise refers to choosing two cards from a thoroughly shuffled deck. Assume that the deck is shuffled after a card is returned to the deck.

If you do not put the first card back in the deck before you draw the next, what is the probability that the first card is a club and the second card is a heart? (Enter your probability as a fraction.)

Answer 1

Since there are 13 clubs in a deck of 52 cards, the probability is 1/4 of a club of the first draw.

I think the problem intends that you should not know the first card before you draw the second.

The probability of a heart on the second draw depends on the suit of the first card drawn.

With probability 1/4 the first card was a heart in which case the probability of another heart is 12/51 and with probability 3/4 the first card was not a heart in which case the probability of a heart is 13/51.

The total probability of a heart on the second draw then is

(1/4)*(12/51) + (3/4)*(13/51) = 51/204 = 1/4

Answer 2

Sampling with replacement:

    13/52 * 13/52 = 1/4*1/4 = 1/16

 

Sampling without replacement:

 

 13/52 * 13/51 = 1/4*13/51 = 13/204 or just over 6%