Sum of 5 or 7: Dice Probability

Date: 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/