Question

Suppose we roll a fair 6 sided die with the numbers [1,6] written on them. After the first die roll we roll the die ? times where ? is the number on the first die roll. The number of points you score is the sum of the face-values on all die rolls (including the first). What is the expected number of points you will score?

Answer 1

There are 6 options when we roll the die first time .

if first roll is 1 . there will be one roll only that have 6 possibilities. (1,2,3,4,5,6) which each have same probability which is 1/6

Pr(first roll = 1) = 1/6 and Expected number on the next 1 roll ,E(k =2) = 1/6 * (1 + 2 + 3 + 4+5+6) = 3.5

so expected sum of die rolls when first roll is 1 is (1 + 3.5 )= 4.5

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Second Case :

when first roll is 2.

then we have to roll the die two times. as these two rolls are independent. So expected value of sum of two rolls are

E(sum of these two rolls are) = 3.5+ 3.5 = 7

Expected sum of rolls when first roll is 2 = 2 + 7= 9

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Case III

WHen first roll is 3

then we have to roll die 3 times As these three rolls are independent expected value of sum of three roll are

E(sum of these 3 rolls are ) = 3.5+3.5+3.5= 10.5

Expected sum of rolls when first roll is 3 = 3 + 10.5 = 13.5

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Case IV

When first roll is 4

then we have to roll die 4 times As these three rolls are independent expected value of sum of 4 roll are

E(sum of these 4 rolls are ) = 3.5+3.5+3.5+3.5=14

Expected sum of rolls when first roll is 4 = 4 + 14= 18

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case V)

when first roll is 5

then we have to roll die 5 times As these three rolls are independent expected value of sum of 5 roll are

E(sum of these 5 rolls are ) = 3.5+3.5+3.5+3.5+3.5=17.5

Expected sum of rolls when first roll is 5 = 5+17.5= 22.5

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when first roll is 6

then we have to roll die 6 times As these three rolls are independent expected value of sum of 6 roll are

E(sum of these 6 rolls are ) = 3.5+3.5+3.5+3.5+3.5+3.5=21

Expected sum of rolls when first roll is 6 = 6+21= 27

so Expected number of points i will score = 1/6 * (4.5 + 9 + 13.5 + 18+22.5+27) = 15.75