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

\[\text{desired prob}=\frac{\text{number of ways of getting desired outcome}}{\text{total number of possible outcomes}}\]
It should be clear that the denominator is \(\dbinom{52}6\), since you are choosing 6 cards at random from a deck of 52.

Of the total 52, only 13 cards are spades, so the number of ways of drawing exactly 2 spades is \(\dbinom{13}2\). The other 4 cards of the hand must be non-spades; there are 39 of these, hence \(\dbinom{39}4\), and so
\[\text{desired prob}=\frac{\dbinom{13}2\dbinom{39}4}{\dbinom{52}6}\]