Increasing Sequence in Dice
Source: A book on probability puzzles
Suppose we play a game with a die where we roll and sum our rolls as long as we keep rolling larger values. For instance, we might roll a sequence like 1-3-4 and then roll a 2, so our sum would be 8. If we roll a 6 first, then we’re through and our sum is 6.
Three questions about this game:
(a) What is the expected value of the sum?
(b) What is the expected value of the number of rolls?
(c) If the game is played with an n-sided die, what happens to the expected number of rolls as n approaches infinity?
Update (Aug 31, 2011)
Solution posted by Gaurav Sinha (chera, IITK 1996 Graduate, Indian Revenue Service) and Siva in comments!