I have been trying to solve this Gambler's Ruin problem in my Statistics class using first principles of Expectation and Variance along with the absorption probability formulas for going broke and winning in a Gambler's Ruin (see below) the probability of going broke for Player 1. the probability of winning for Player 1.
The problem is as follows: "Players A and B decide to gamble against each other by rolling a fair dice until one of them runs out of money. At the beginning, A has 40 dollars and B has 50. When the dice lands on 1 or 3 or 4 or 6, A wins 1 dollar and B loses 1. When the dice lands on 2 or 5, B wins 1 dollar and A loses 1. Find the variance of B's final wealth."
I have been trying to find Var[X], where X = B's final wealth, using Var[X] = E[X^2] - E[X]^2. The 2 outcomes for X are 0 or 90. However, using the formulas, I get that the probability of B going broke is 1 and the probability of B winning is 0, leading to E[X] = 1*0 + 0*90 = 0 and leading to a variance of 0.
This just doesn't feel right to me. How would you guys go about solving this problem? Thank you for your help!