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

Let S be the sum when 5 fair dice are rolled. Find the pgf (probablity Generating Functions) of S???

and describe how you can use it to find the probability P(S = 20).???

please help me

Solution

IF 5 dice are thrown totals can be 5,6,7.....30

No of outcomes = 6^5 =7776

Thus the prob function is listed above

P(s=20) = 0.083719

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Working note:

For 5 (1,1,1,1,1)

     6 (2,1,1,1,1) 2 can be any of the 5 places

     7 (2,2,1,1,1) Or (3,1,1,1,1) (shuffled)

and so on