Problem

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 of .6.

Find the probability that he wins 8 dollars before losing all of his money if

Example A

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

Strategy

p=0.4 based on the probability of wins q=0.6 probability of failure on an individual trial M=8 based 8 dollars needed to get out on bail Smith= 1 based on the dollar he has

Calcuate the numerator

#divide the probability of failure by the probability of wins
0.6/0.4
## [1] 1.5
#multiply 1.5 to the power of 1 and subtract Smith's dollar bet
1.5^1-1
## [1] 0.5

Calculate the denominator

#multiplty 1.5 to the power of 8 which represents the 8 dollars needed for Smith to get out of jail and subtract 1
1.5^8-1
## [1] 24.62891

Divide

cat("Smith's Timid Strategy","\n")
## Smith's Timid Strategy
0.5/24.62891
## [1] 0.02030135

Example B

  1. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy)

As we look at this situation we see that in order for Smith to reach 8 dollars he has to be aggressive in his betting. Therefore, Smith must match his winnings at every turn plus win.

Scenario

Bet #1: Smith bets 1 dollar and wins and matches or loses first bet; if he wins he can continue tieh 2 dollars, if he loses game over.

Bet #2: Smith bets 2 dollars and wins turning his winnings to 4 dollars or he loses.

Bet #3: This is where Smith can turn the 4 dollar bet into the 8 dollars needed to get out of jail.

Simply put the probability of 0.4 can be raised to the power of 3 because if Smith loses between his first three bets he will remain in jail.

cat("Smith's Bold Strategy","\n")
## Smith's Bold Strategy
0.4^3
## [1] 0.064

Example C

  1. which strategy gives Smith the better chance of getting out of jail?

With the chances of acquiring the required 8 dollars seeming impossible with the timid strategy it is best for Smith to go bold. Go hard or go home!