Probability and Statistics

Question

3 cards are drawn from a standard deck without replacement. What is the probability that at least one of the cards drawn is a spade? Express your answer as a fraction or a decimal rounded to four decimal places.

Answer 1

Hi Gina,

The quickest way to answer this questions is to answer its complement. You have been asked about the probability of seeing at least one spade in 3 cards. The complement of this, is getting no spades. Once we get the probability of "no spades", then we can subtract that value from 1 to get the probability of "at least on spade". The probability of "no spades" means each card we draw is not a spade. Since we are doing this without replacement, this means the number of cards in the deck decrease as well as the number of non-spade cards as well. We start with 39 non-spade cards out of 52 and go down from there. So we will get (39/52)*(38/51)*(37/50)=54834/132600 as the probability of getting "no spades". So, using the complement rule, you will get 1- 54834/132600=0.5865 as the probability of observing at least one spade.

Answer 2

Well probability that none are spades is as follows :

We can choose the cards in 52 C 3 ways

Of these we can choose 3 cards that are not spades in 39 C 3 ways

So p (not a spade) = 39 C 3 / 52 C 3

So p (at least one spade ) = 1 - ( 39 C 3 / 52 C 3)