Data 605 Assignment 10

Problem 1

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).

p <- 0.4
q <- 0.6
r <- q/p


for (i in 0:8){
P <- round((1-r^i)/(1-r^8),4)
print (P)
}
## [1] 0
## [1] 0.0203
## [1] 0.0508
## [1] 0.0964
## [1] 0.1649
## [1] 0.2677
## [1] 0.4219
## [1] 0.6531
## [1] 1

(B) He bets, each time, as much as possible but not more than necessary to

bring his fortune up to 8 dollars (bold strategy).

b <- matrix(c(1,0.6,0.6,0.6,0,0,0,0,0,0,0,0.4,0,0,0,0,0,0.4,0,0,0,0,0,0.4,1),nrow = 5, byrow=FALSE)
b
##      [,1] [,2] [,3] [,4] [,5]
## [1,]  1.0    0  0.0  0.0  0.0
## [2,]  0.6    0  0.4  0.0  0.0
## [3,]  0.6    0  0.0  0.4  0.0
## [4,]  0.6    0  0.0  0.0  0.4
## [5,]  0.0    0  0.0  0.0  1.0
b1 <- matrix(c(0,1,0,0,0),nrow=1)
b1
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    1    0    0    0
b2 <- b1 %*% b
b2
##      [,1] [,2] [,3] [,4] [,5]
## [1,]  0.6    0  0.4    0    0
b3 <- b2 %*% b
b4 <- b3 %*% b
b4
##       [,1] [,2] [,3] [,4]  [,5]
## [1,] 0.936    0    0    0 0.064

The probability is 0.064

(C) Which strategy gives Smith the better chance of getting out of jail?

The probability of bold strategy is better than timid strategy to get out of jail.