Consider the following game: A player throws a fair die repeatedly until he rolls a 2, 3, 4, 5,or 6.
In other words, the player continues to throw the die as long as he rolls Is. When he
rolls a "non-1", he stops. (Show your work and circle your final answers).
a) What is the probability that the player tosses the die exactly three times?
b) What is the expected number of rolls needed to obtain the first non-1?
c) If he rolls a non-1 on the first throw, the player is paid $1. Otherwise, the payoff is doubled
for each 1 that the player rolls before rolling a non-1. Thus, the player is paid $2 if he rolls a 1
followed by a non-1; $4 if he rolls two Is followed by a non-1; $8 if he rolls three Is followed by
a non-1; etc. In general, if we let Y be the number of throws needed to obtain the first non-1,
then the player rolls (Y-1) Is before rolling his first non-1, and he is paid 2^{Y-1} dollars. What
is the expected amount paid to the player?
