Question

Andrea and raleigh are each rolling a fair, six-sided die. they roll their dice simultaneously, individually keeping a sum until someone reaches 100; whoever reaches 100 first wins. (if they reach 100 on the same roll, it's a tie.) andrea's die has sides 1, 2, 3, 4, 5, and 6. raleigh's has sides 1, 1, 1, 6, 6, and 6. who is more likely to win?

Answer 1

Let  An be the likely number of rolls for Andrea to arrive at 100 or more, given that she is  n away from the target of  100 (which means her current total is 100 - n ). Then we get the recurrence relation

An = 1 + (1/6) (An-1 + An-2 + An-3 + An-4 + An-5 + An – 6)

and the initial conditions A0 = A-1 = A-2 = A-3 = A-4 = A-5 = 0. This results in A100 = 29.0476 as the amount of expected rolls that Andrea has to make.

Now let  Rn to be expected amount of rolls for Raleigh to arrive at 100 or more, given than he is also  n away from the target of 100. Now the recurrence relation is:

Rn = 1 + (1/2) (Rn-1 + Rn-6)

with the initial conditions R0 = R-1 = R-2 = R-3 = R-4 = R-5 = 0. This gives us R100 = 29.1837 as the expected amount of rolls for Raleigh.

 

Therefore Andrea has to make less number of rolls thus she is expected to win.

Winner: Andrea