Sum of 5 or 7: Dice ProbabilityDate: 09/11/2002 at 00:54:36 From: Ping Fu Subject: Probability of dice A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 will occur first. I have no idea where to start on this question. Thanks. Date: 09/11/2002 at 07:49:00 From: Doctor Anthony Subject: Re: Probability of dice At each throw we have: Probability of 5 = 4/36 = 1/9 Probability of 7 = 6/36 = 1/6 Probability of no result = 1 - (1/9 + 1/6) = 13/18 Method (1) - Difference Equation: Let p = the probability that we will roll a total of 5 before we roll a total of 7. If we roll the two dice, we can win with probability 1/9, lose with probability 1/6, or return to start with probability 13/18, and so p = 1.(1/9) + 0.(1/6) + p(13/18) p[1 - 13/18] = 1/9 p(5/18) = 1/9 p = 2/5 Method (2) - Summing an infinite series: p = 1/9 + (13/18)(1/9) + (13/18)^2(1/9) + (13/18)^3(1/9) + ... = (1/9)[1 + 13/18 + (13/18)^2 + (13/18)^3 + ...... ] This is a GP with common ratio 13/18 = (1/9)/[1 -13/18] = (1/9)/(5/18) = 2/5 And so the probability of a 5 before a 7 is 2/5. We could have expected this result, since the ratio of the probabilities is P(5) : P(7) ------------- 2/18 : 3/18 = 2 : 3 2 2 and probability of 5 before 7 is ----- = --- 2+3 5 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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