Five cards are to be drawn in succession and without replacement from an ordinary deck of playing cards.

Question: Five cards are to be drawn in succession and without replacement from an ordinary deck of playing cards.

(a) What is the probability that there will be no ace among the five cards drawn?
(b) What is the probability that the first three cards are aces and the last two cards are kings?
(c) What is the probability that only the first three cards are aces?
(d) What is the probability that an ace will appear only on the fifth draw?

Solution:

(a) The probability that there will be no ace among the five cards:
p = 48/52 x 47/51 x 46/50 x45/49 x 44/48 = 205476480/311875200.

(b) The probability that the first three cards are aces and the last two cards are kings :
p = 4/52 x 3/51 x 2/50 x4/49 x 3/48 = 288/311875200.

(c) The probability that only the first three cards are aces:
p = 4/52 x 3/51 x 2/50 x48/49 x 47/48 = 54144/311875200.

(d) The probability that an ace will appear only on the fifth draw:
p = 48/52 x 47/51 x 46/50 x45/49 x 4/48 = 18679680/31875200.

The multiplication theorem will hold good only if the events belong to the same set. In order to show the importance of this fact, Moroney in his book “facts from Figures” gives an interesting example. He observes, “Consider the case of a man who demands the simultaneous occurrence of many virtues of an unrelated nature in his young lady. Let us suppose that he insists on a Grecian nose, platinum-blonde hair, eyes of odd colours – one blue, one brown, and finally a first class knowledge of statistics. What is the probability that the first lady he meets in the street will put ideas of marriage into his head? It is difficult to apply multiplication theorem in this case, because events do not belong to the same set.