Consider rolling a regular (fair) six-sided die, with sides numbered 1 through 6 in the usual fashion. For some fixed positive integer S, let RS be the number of times you roll the die until the sum of the rolls is greater than or equal to S.
(a) If S=2, find the probability distribution of R2
and show that E[R2]=7/6.
(b) If S=3, find the probability distribution of R3
and show that E[R3]=(7/6)2.
(c) If S=6, find the probability distribution of R6 and show that E[R6]=(7/6)5.
(d) Conjecture: For S a positive
integer, E[RS]=(7/6)S-1. Prove or disprove.