Let S be the sum when 5 fair dice are rolled Find the pgf of

Let S be the sum when 5 fair dice are rolled. Find the pgf of S and describe how you can use it to find the probability P(S = 20).

Solution

When 5 fair dice are rolled outcome can be 5, 6, 7,......30

The total outcomes will be 6⁵ = 7776

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Favourable outcomes for 5 =1

                                   30 =1

For 6 is 4 dice 1 and any one dice 2 i.e. no of ways = 5c1 =5

For 7, 4 dice 1 and any one 3 Or 2 dice 2, 3 dice 1

   no of ways. = 5C1+5c2 = 15.

For 8, it would be 4 dice1, 1 die 4, 2 dice 1, 2 dice 3, or 1 die 3 and 1 die 2, 3 die 1 or 3 die 2, 2 die 1 or

=35

Thus sums will be like the one listed below:

For 8,

         Hence prob for (s=20) = 0.0837186