Question

5 cards are drawn at random from a standard deck.
a)Find the probability that all the cards are face cards.
b)Find the probability that all the cards are even

Answer 1

A) Out of the 52 cards in the deck, 12 are face cards (16 if you count aces). The probability of drawing the first face card is then 12/52 (16/52 if youcount aces). After that there is one less card in the deck and 1 less face card so the probability of drawing the second face card is 11/51 (15/51 if youcount aces). That pattern continues for the 3rd, 4th and 5th face cards. Sinces all of these events have to occur in sequence, the probability of thathappening is the product of each individual event. That will be:

(12/52)(11/51)(10/50)(9/49)(8/48) = 33/108290

Or if we count aces:

(16/52)(15/51)(14/50)(13/49)(12/48) = 1/595

B) This problem is very similar to the first. In the deck there are 2's, 4's, 6's, 8's and 10's in each suit, so there ar 4*(5) = 20 even cards. Theprobability of drawing the first one is (20/52) the second (19/51) the third (18/50) and so on. The product will look like this:
(20/52)(19/51)(18/50)(17/49)(16/48) = 19/3185

Answer 2

A) Out of the 52 cards in the deck, 12 are face cards (16 if you count aces). The probability of drawing the first face card is then 12/52 (16/52 if youcount aces). After that there is one less card in the deck and 1 less face card so the probability of drawing the second face card is 11/51 (15/51 if youcount aces). That pattern continues for the 3rd, 4th and 5th face cards. Sinces all of these events have to occur in sequence, the probability of thathappening is the product of each individual event. That will be:

(12/52)(11/51)(10/50)(9/49)(8/48) = 33/108290

Or if we count aces:

(16/52)(15/51)(14/50)(13/49)(12/48) = 1/595

B) This problem is very similar to the first. In the deck there are 2's, 4's, 6's, 8's and 10's in each suit, so there ar 4*(5) = 20 even cards. Theprobability of drawing the first one is (20/52) the second (19/51) the third (18/50) and so on. The product will look like this:
(20/52)(19/51)(18/50)(17/49)(16/48) = 19/3185