# Actuary Probability problem

• Jan 11th 2013, 11:27 AM
feefers
Actuary Probability problem
The game of bridge is played by four players: north, south, east and west.
Each of these players receive 13 cards.
(a) What is the probability that one of the players receives all 13 spades?

I am having serious trouble with this one, the answer in the back of the book is 6.3  10^-12 and I cant seem to match it.

At first I thought it may be C(13,13)/C(52,13) but thats considering only one player.

Then I thought maybe C(13,13)*C(39,13)*C(26,13)*(13,13)/C(52,52) but this does not work either.

Help!!
• Jan 11th 2013, 12:33 PM
Plato
Re: Actuary Probability problem
Quote:

Originally Posted by feefers
The game of bridge is played by four players: north, south, east and west.
Each of these players receive 13 cards.
(a) What is the probability that one of the players receives all 13 spades?

There are $\displaystyle \frac{52!}{(13!)^4}$ ways to deal a bridge game.

There are $\displaystyle 4\cdot \frac{39!}{(13!)^3}$ ways for one player to receive all 13 spades.
• Jan 12th 2013, 05:10 AM
awkward
Re: Actuary Probability problem
Another way to look at the problem:

There are $\displaystyle N = \binom{52}{13}$ possible bridge hands. Only one of these has all the spades. So the probability that one of the players has all the spades is $\displaystyle 4 / N$.