Should You Play?

Date: 08/07/97 at 22:29:16
From: Connie
Subject: Face card probability

A game consists of three cuts with a deck of 52 cards. You win $1 if a 
face card turns up on at least one cut, but you lose $1 if a face card 
does not turn up. Should you play? Why?

Thanks,
Connie

Date: 08/08/97 at 05:03:24
From: Doctor Anthony
Subject: Re: Face card probability

A game is considered fair if the expectation is zero.  That is, if in 
a long series of games you would expect to break even.

The definition of 'expectation' if x1 occurs with probability p1, 
x2 occurs with probability p2, and so on, is:

   E(x) = p1.x1 + p2.x2 + p3.x3 + ...... + pn.xn

To find the probabilities of at least one face card in 3 trials, 
consider instead the probability of no face card in 3 trials, and 
subtract this probability from 1.

Probability of no face card in three trials = (9/13)^3 = .331816

Probability of at least one face card in 3 trials = 1 - (9/13)^3

                                                  = 0.66818
Your expectation = 1 x .66818 - 1 x .331816

                 = 0.33637

With a positive expectation you should certainly play.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/