Coin-Tossing GameDate: 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/ |
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