We wanted to find the probability, x, that Bob wins a game of Stupid Spin at Scam Casino. The probability Bob gets a point is a, otherwise Scam Casino gets one. The first to b points wins. First, the maximum spins in a round is 2b – 1, where 1 player wins by just 1 point. If we always play this many spins, you will always win at b points. We can simplify the problem to whomever has more points after 2b – 1 spins. Bob wins if the Casino has 0 to b-1 points. This gives us the first part of the equation. The probability of any given score for the Casino is equal to the number of ways that score can happen times the probability of it happening. The number of ways is easily done with the choose function. The probability of that is a to the power of the number of points scored by Bob, part three, multiplied by 1-a to the power of the number of points scored by the casino, the final part. We combine these 4 parts to get our answer, the equation below.
\sum_{k=0}^{b-1} {2b-1 \choose k}a^{2b-k-1}(1-a)^{k}