Coin-Tossing Game

Date: 05/09/2001 at 02:40:56
From: Ashutosh
Subject: Probability

Three men toss a coin in succession for a prize to be given to the 
person who first obtains a head. What is each man's chance of winning 
the prize?

Date: 05/09/2001 at 03:44:45
From: Doctor Floor
Subject: Re: Probability

Hi, Ashutosh,

Thanks for writing.

The probability that the first person will win the prize on the first 
toss is clearly 1/2.

For the second person, the probability that he will win on the first 
toss is 1/2 * 1/2 = 1/4, because to get the toss, first the first 
person has to lose (this gives the first 1/2) and then he has to 
obtain a head.

For the third person, the probability that he will win on the first 
toss is 1/4 * 1/2 = 1/8, because the probability that he will get a 
toss is 1/4 (left over after winning probabilities 1/2 and 1/4 from 
the first two persons).

So we see that for the first toss, the probability that the first 
person will win is 4 times the probability that the third person will 
win, and the probability that the second person will win is 2 times 
the probability that the third person will win. The probabilities have 
a (combined) ratio of 4:2:1.

This ratio is for the first toss, but it is not difficult to imagine 
that this is also true for the second toss, and also for the third and 
later tosses.

So the ratio of probabilities for the total number of tosses needed is 
again 4:2:1. Since the sum of the probabilities is 1, we conclude, 
using that 4 + 2 + 1 = 7:

     probability first person wins  = 4/7
     probability second person wins = 2/7
     probability third person wins  = 1/7

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/