Question: Yahtzee is a game where you roll 5 dice and get different points depending on the values of the dice. A Yahtzee is scored when someone rolls five-of-a-kind on their turn. A turn consists of three rolls. On the first roll, you must roll a 5 dice. On the second and third rolls, you may choose to roll any of the five dice.

(a) Assume you roll just one die. What is the probability that you will roll a 2 within your first three rolls? (Hint: You would stop rolling if you got a 2 on your first roll.)

(b) Now assume you roll all five dice. What is the probability that on your turn you roll a Yahtzee consisting of all 2s? (Hint: use your results from part (a).)

(c) The probability of rolling a Yahtzee of any combination on one turn is 0.046. (Recall that each turn consists of three rolls.)

i. What is the probability of rolling your first Yahtzee on the ₅th turn?

ii. I will give you 5 turns to roll a Yahtzee. If you roll a Yahtzee you win and stop rolling. What is the probability you win?

iii. What is the expected number of turns before rolling a Yahtzee?