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

Solution:

Let’s use the formula from Chapter 12.2 “GAMBLER’S RUIN”.

```
p <- 0.4
q <- 0.6
z <- 1
M <- 8
prob <- ((q/p)^z - 1)/((q/p)^M - 1)
prob
```

`## [1] 0.02030135`

Solution:

Let’s set up a 5x5 matrix for the Markov chain for this scenatio. We have the possible outcomes of 0,1,2,4,8 and there is what the probabilities look like:

```
tmatrix <- matrix(c(1,0,0,0,0,0.6,0,0.4,0,0,0.6,0,0,0.4,0, 0.6, 0, 0, 0, 0.4,0,0,0,0,1), nrow=5, byrow=TRUE)
tmatrix
```

```
## [,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
```

```
init <- c(0,1,0,0,0)
p2<-init%*%tmatrix
p2
```

```
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6 0 0.4 0 0
```

```
p3<- p2 %*% tmatrix
p3
```

```
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.84 0 0 0.16 0
```

```
p4<- p3 %*% tmatrix
p4
```

```
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.936 0 0 0 0.064
```

Solution:

The bold strategy gives a higher chance of success -0.064 - than the timid strategy - 0.02 (I wonder if there is a life lesson somewhere in this problem - not about gambling though).