If 6 cards are drawn at random from a standard deck of 52 cards, what is the probability that exactly 2 of the cards are spades
A. 0.038
B. 0.200
C. 0.315
D. 0.465
E. 0.747

Question

If 6 cards are drawn at random from a standard deck of 52 cards, what is the probability that exactly 2 of the cards are spades
A. 0.038
B. 0.200
C. 0.315
D. 0.465
E. 0.747

kirbykirby · Answer

Well the sample space consists of all possible cards drawn, regardless of what they are. Also, the order of the cards in your draw pile doesn't matter... which should be a hint to use combinations ;)

So the sample space consists of all 6 possible cards from 52 than you can choose, so $\large {52\choose 6} $

Now, 13 cards out of 52 are spades. And you need to get 2 spades out of 13 from the deck, so $\large {13\choose 2} $. Now, the 4 remaining cards can't be spades, so you must choose 4 non-spades from all the remaining non-spade cards (which is 52 - 13 = 39), so you can get 4 non-spades in $\large {39\choose 4} $