QOTW 5

Pg 13, E4

In raquetball, a player continues to serve as long as she is winning; a point is scored only when a player is serving and wins the volley. The first player to win 21 points wins the game. Assume that you serve first and have a probability 0.6 of winning a volley when you serve and probability 0.5 when your opponent serves. Estimate, by simulation, the probability that you will win a game.

Raquetball <- function() {
    #Tracks who the current server, used to assign points
    Serving <- 1
    
    #Probabilities 
    Prob_win_serve_1st <- 0.6
    Prob_win_serve_2nd <- 0.5
    
    #1 = I win. 0 = Opponent wins
    outcomes <- c(1,0)  # 1, 0 will be used to count how many times I win in n games
    
    #tracking points
    Score_mine <- 0
    Score_Opponent <- 0
    
    #game loops until someone reaches 21 points
    while (Score_mine < 21 & Score_Opponent < 21) {
      
      #If opponent serves, use P = 0.5
      if (Serving == 0) {
        #probabilities <- c(Prob_win_serve_2nd, 1-Prob_win_serve_2nd)
        who_won <- sample(outcomes, 1, prob =  c(0.5,0.5))
        #If opponent wins, they get a point, otherwise I serve
        if (who_won == 0) {
          Score_Opponent <- Score_Opponent + 1  
        } else {
          Serving <- 1
        }
        
      } else {
        #If I serve, use P = 0.6
        #probabilities <- c(Prob_win_serve_1st, 1- Prob_win_serve_1st)
        who_won <- sample(outcomes, 1, prob =  c(0.6,0.4))
        #If I win,add a point to my total, otherwise opponent serves
         if (who_won == 1) {
          Score_mine <- Score_mine + 1  
        } else {
          Serving <- 0
        }
      }
      
    }
    
    if (Score_mine == 21) {
      return(1)
    }
    else {
      return(0) 
    }
}

Let’s see the chance of winning a game in 5000 games

count <- c()
simulations <- 5000

for (i in seq(simulations)) {
  game <- Raquetball()
  count <- c(count, game)
}

Prob_win <- sum(count) / length(count)
cat("The probability to win in ", simulations, " games is: ", Prob_win )

## The probability to win in  5000  games is:  0.83