Cmput 455 Lecture 15, details for slide "Wrong Choice in Bandits", code binomial-select.py
Detailed explanation for the first result computed by the program and shown in binomial-select-experiment.txt 

p1 = 0.9, p2 = 0.2
Both have 1 simulations. Prob. of wrong arm choice 0.15

Why?

4 cases, each of the 2 sim. can be either win or loss. Compute the probability of each case.
50/50 p1 win and p2 win: 0.9*0.2 = 0.18
GOOD p1 win and p2 loss: 0.9*0.8 = 0.72
BAD p1 loss and p2 win: 0.1*0.2 = 0.02
50/50 p1 loss, p2 loss: 0.1*0.8 = 0.08

50/50: if both arms have the same number of wins, we choose randomly
GOOD: the better arm (p1 here) has more wins
BAD: the worse arm (p2 here) has more wins

p(make the right choice) = 0.72 + 0.5(0.18 + 0.08) = 0.72 + 0.13 = 0.85
p(make wrong choice) = 0.02 + 0.5(0.18 + 0.08) = 0.02 + 0.13 = 0.15