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?



0.038



0.200



0.315



0.465



0.747

Answer 1

So, there are spades, hearts, diamonds and clubs in your deck. The deck is divided into 4 of these suits. So, there are going to be 52/4 = 13 spades in your deck.

Note that the order of the cards don't matter.

Now, for the sample space: you are picking 6 cards out of 52, which means:
\[ {52 \choose 6}\]

Now for the event, you need to pick exactly 2 spades. That means that the remaining 4 cards you pick cannot be spades!!

So, choosing 2 spades out of the 13 spades you have in your deck is represented as:
\[ {13 \choose 2}\]Now, the other 4 cards have to be picked from cards that are NOT spades. Since you must pick from the non-spades cards, there are 52 - 13 = 39 cards that aren't spades.

So, the 4 remaining cards are chosen from the 39 non-spade cards:
\[ {39 \choose 4}\].

So the probability is:
\[\large \frac{{13\choose 2}{39\choose 4}}{{52 \choose 6}}\].

If you are familiar with the hypergeometric distribution, this is exactly what it is!

Answer 2

(one trick to see if you got it correct: the two top number in the numerator, i.e. the 13 and 39, must add to 52,
and the two bottom numbers, the 2 and 4, must add to 6)