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 (a) he bets 1 dollar each time (timid strategy).

Using this equation to find the probability

q_z = 

p <- 0.4  
q <- 0.6  
z <- 1    
m <- 8

q_z <- ((q/p)^z - 1) / ((q/p)^m - 1)
print(q_z)
## [1] 0.02030135
  1. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
p^3
## [1] 0.064
  1. Which strategy gives Smith the better chance of getting out of jail?

When we compare the two strategies, the bold one clearly gives Smith a much better chance of success. With a probability of about 6.4%, the bold strategy is significantly more effective than the timid one, which only gives a probability of roughly 2.03%. This shows that betting more aggressively increases Smith’s likelihood of reaching 8 dollars before losing all his money. Therefore, in this situation, choosing the bold strategy is the smarter choice for Smith as it greatly improves his chances of getting out of jail.