Assignment 10

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

1. he bets 1 dollar each time (timid strategy).
2. he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
3. Which strategy gives Smith the better chance of getting out of jail?

Ans.

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

transitionMatrix = rbind(# $1 $2 $3 $4 $5 $6 $7 $0 $8
                         c(0 ,.4,0 ,0 ,0 ,0 ,0 ,.6,0 ), # $1
                         c(.6,0,.4 ,0 ,0 ,0 ,0 ,0 ,0 ), # $2
                         c(0 ,.6,0 ,.4,0 ,0 ,0 ,0 ,0 ), # $3
                         c(0 ,0 ,.6,0 ,.4,0 ,0 ,0 ,0 ), # $4
                         c(0 ,0 ,0 ,.6,0 ,.4,0 ,0 ,0 ), # $5
                         c(0 ,0 ,0 ,0 ,.6,0 ,.4,0 ,0 ), # $6
                         c(0 ,0 ,0 ,0 ,0 ,.6,0 ,0 ,.4), # $7
                         c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ), # $0
                         c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 )  # $8
                         )

Q = transitionMatrix[1:7, 1:7]

I = diag(7)

N = inv(I-Q) #  Analogous to geometric series N = sum( Q^n ) = 1/( 1-q)

R = transitionMatrix[1:7, 8:9]
           
B = N %*% R #Absorption Probability Matrix

timiProb = B[1,2]

timiProb

## [1] 0.02030135

The probability that he wins 8 dollars before losing all of his money if he bets 1 dollar each time is 0.0203013.

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

transitionMatrix = rbind(# $1 $2 $3 $4 $5 $6 $7 $0 $8
                         c(0 ,.4,0 ,0 ,0 ,0 ,0 ,.6,0 ), # $1
                         c(0 ,0 ,0 ,.4,0 ,0 ,0 ,.6,0 ), # $2
                         c(0 ,0 ,0 ,0 ,0 ,.4,0 ,.6,0 ), # $3
                         c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,.6,.4), # $4
                         c(0 ,.6,0 ,0 ,0 ,0 ,0 ,0 ,.4), # $5
                         c(0 ,0 ,0 ,.6,0 ,0 ,0 ,0 ,.4), # $6
                         c(0 ,0 ,0 ,0 ,0 ,.6,0 ,0 ,.4), # $7
                         c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ), # $0
                         c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 )  # $8
                         )

Q = transitionMatrix[1:7,1:7]

I = diag(7)

N = inv(I-Q)

R = transitionMatrix[1:7,8:9]
          
B = N %*% R

boldProb = B[1,2]

boldProb

## [1] 0.064

The probability that he wins 8 dollars before losing all of his money if he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars is 0.064.

3. Which strategy gives Smith the better chance of getting out of jail?

Ans. The bold strategy has the higher probability of absorption in state which gives Smith the bettter chance of getting out of jail.