Lionel wants to play the crane machine arcade game (the one with the mechanical claw) to win a stuffed animal. He is good at the game, so for each time he plays, there is a 10% chance that he will win a stuffed animal, regardless of the previous results.
a) Let A be the number of rounds until Lionel wins his first stuffed animal. What is the distribution and parameters of A? Also, find the expected value and variance of A.
b) What is the probability that it takes exactly 10 games for Lionel to win a stuffed animal?
c) What is the probability that it takes at least 7 games until Lionel wins a stuffed animal?
d) Lionel has already played 5 games and has not won a stuffed animal, what is the probability that it takes no more than 12 games altogether until he wins a stuffed animal?
e) Suppose that it costs $1 to play a game and the stuffed animal in the toy store costs $15. On average is it worthwhile for Lionel to play the crane machine game to get the stuffed animal instead of buying one in the toy store? Justify your answer mathematically.