Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability .6.
Find the probability that he wins 8 dollars before losing all of his money if
he bets 1 dollar each time (timid strategy).
p = 0.4
q = 0.6
r = q/p
s=1 #start
m=8 #max
p_win = (1-(r^s))/(1-(r^m))
paste0('Probability is :', round(p_win ,4) )## [1] "Probability is :0.0203"he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
==> This can be solved with a Binomial Distribution.
paste0('Probability is :', round(dbinom(3,3,0.4) ,4) )## [1] "Probability is :0.064"Which strategy gives Smith the better chance of getting out of jail?
==> Smith triples his probability of winning if he uses the bold strategy; that gives Smith the better chance of getting out of jail.