aez-notes

Mr. Brown always bets a dollar on the number 13 at roulette against the advice of Kind Friend. To help cure Mr. Brown of playing roulette, Kind Friend always bets Brown \$20 at even money that Brown will be behind at the need of 36 plays. How is the cure working?

(Most American roulette wheels have 38 equally likely numbers. If the player's number comes up, he is paid 35 times his stake and gets his original stake back; otherwise he loses his stake)

First consider the roulette, the probability of losing the stake is \(37/38\) otherwise, \(35\) is won. This allows us to compute the average result. Since the friendly bet is won so long as Brown gets at least one spin, this allows us to compute the expected result. Putting these together we find that the cure is not working because on average Brown will win.

load(distrib)$
roulette_expected_outcome : 36*((-1) *(37/38) + (35)*(1/38)), numer;
f(x) := pdf_binomial(x,36,1/38);
friendly_expected_outcome : 20 * (1 - f(0)) + (-20) * f(0), numer;
print(friendly_expected_outcome + roulette_expected_outcome);

Mr. Brown can expect to win 2.79.