Expert: Josh - 3/3/2009

Question
Hey -  I understand how to do a lot of these problems, but this one has been giving me trouble.

Consider a standard deck of cards. You will draw three cards.  What is the probability of of choosing all three cards from the same suit?

If you can help with this, I would very much appreciate it. Thanks

Answer
Hi Kelly

A standard deck contains 52 cards. We have 13 cards {2,3,4,5,6,7,8,9,J,Q,K,A} for spade, heart, club and diamond.

Pick 1:
The first card must be one of spade, heart, club or diamond. There are 4 distinct possibilities. For the purpose of analysis, it doesn't matter which one we pick. So, let's say we picked diamond first up.

Pick 2:
If we pick diamond, we will be left with 12 cards out of 51 in the diamond suit. We no longer have the full deck as one card (of diamond) has been removed from it.

The probability for the second card to be diamond is 12/51.

Pick 3:
If we pick diamond again, we will be left with 11 cards out of 50 in the diamond suit. (Now, we have two cards removed from the full deck)

So, the probability for the third card to be diamond is 11/50.

To get three diamonds in a row, this sequence of events must take place independently, one after the other. For this reason, we multiply these probabilities to get (12/51)x(11/50). This figure only accounts for "3 consecutive cards of diamond". Since the same argument holds for the other three suits (club, heart and spade), overall, we have a chance of 4x(12/51)x(11/50) of getting three cards of the same suit.