Consider a game where 3 contestants each roll an n-sided die. On the first play of the game, each will roll a 3-sided die. On the next play of the game, each contestant will roll a 4-sided die. On each subsequent play, the number of sides on the die will increase by 1.
a) Suppose on any play of the game, the contestants each win a car if the three resulting rolls form a sequence. If the contestants keep playing forever, will the contestants win a finite or infinite number of cars?
Chance of winning on the nth play?
Hint: there will be n+2 sides on each die on the nth play. There are n possible sequences: ranging from {1,2,3} to {n,n+1,n+2}. Find the chance of a specific sequence, like {1,2,3}, and then extend to the chance of obtaining any sequence in general, realizing each sequence has the same chance.
According to the Borel-Cantelli Lemma, the contestants will win a finite number of cars.
Select the two correct answers. Justify answer.
b) Suppose on any play of the game, the contestants each win $5 if any of the 3-die match (a pair or triple is needed to win). If the contestants keep playing forever, will the contestants win a finite or infinite amount of money?
Chance of winning on the nth play?
Hint: there will be n+2 sides on each die on the nth play. The chance of losing requires player 2 to land different from player 1, and then player 3 to land different from both player 1 and player 2.
According to the Borel-Cantelli Lemma, the contestants will win an infinite amount of money.
Select the two correct answers. Justify answer.