Grando Discussion Week 10

Chapter 11.2 Exercise 15

Question:

Consider the game of tennis when deuce is reached. If a player wins the next point, he has advantage. On the following point, he either wins the game or the game returns to deuce. Assume that for any point, player A has probability .6 of winning the point and player B has probability .4 of winning the point.

1. Set this up as a Markov chain with state 1: A wins; 2: B wins; 3: advantage A; 4: deuce; 5: advantage B.

Answer:

The transition matrix for this problem is:

(tm <- matrix(c(1,     0,     0,    0,    0,
                0,     1,     0,    0,    0,
                0.6,   0,     0,    0.4,  0,
                0,     0,     0.6,  0,    0.4,
                0,     0.4,   0,    0.6,  0
                ), ncol = 5, byrow = TRUE))

##      [,1] [,2] [,3] [,4] [,5]
## [1,]  1.0  0.0  0.0  0.0  0.0
## [2,]  0.0  1.0  0.0  0.0  0.0
## [3,]  0.6  0.0  0.0  0.4  0.0
## [4,]  0.0  0.0  0.6  0.0  0.4
## [5,]  0.0  0.4  0.0  0.6  0.0

Question:

2. Find the absorption probabilities.

Answer:

From this week’s reading, we found that the probability for each initial state can be found by computing B = NR…

So, R is:

(rm <- matrix(c(0.6, 0,    0,
                0,   0,    0.4
                ), nrow = 3, byrow = FALSE))

##      [,1] [,2]
## [1,]  0.6  0.0
## [2,]  0.0  0.0
## [3,]  0.0  0.4

and \(N = {\left( I - Q \right)}^{-1}\):

qm <- matrix(c(0,    0.4,  0,
               0.6,  0,    0.4,
               0,    0.6,  0
               ), ncol = 3, byrow = TRUE)
(nm <- round(solve(diag(3) - qm), 4))

##        [,1]   [,2]   [,3]
## [1,] 1.4615 0.7692 0.3077
## [2,] 1.1538 1.9231 0.7692
## [3,] 0.6923 1.1538 1.4615

So, NR, or B, is:

(bm <- round(nm %*% rm, 3))

##       [,1]  [,2]
## [1,] 0.877 0.123
## [2,] 0.692 0.308
## [3,] 0.415 0.585

and t is:

(tm <- round(nm %*% matrix(rep(1,3), ncol = 1), 4))

##        [,1]
## [1,] 2.5384
## [2,] 3.8461
## [3,] 3.3076

Question:

3. At deuce, find the expected duration of the game and the probability that B will win.

Answer:

The expected duration at deuce, which is state 4, is 3.8461 games and the probability that B will win is 30.8%.