Week 5 Discussion - Simulation of Discrete Probabilities

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 .6 of winning a volley when you serve and probability .5 when your opponent serves. Estimate, by simulation, the probability that you will win a game.

To answer this question, I’ll write a function called raquetball that simulates a single game, based on the probabilities given.

raquetball <- function() {
    #tracks who the current server is so we know who to give a point to 
    who_is_serving <- 1
    
    #probabilities 
    my_win_prob_while_serving <- 0.6
    my_win_prob_opponent_serving <- 0.5
    
    #1 = I win, 0 = You win
    outcomes <- c(1,0) 
    
    #tracking points
    my_points <- 0
    her_points <- 0
    
    #game continues (loops) until someone reaches 21 points
    while (my_points < 21 & her_points < 21) {
      
      #if opponent is serving, use 0.5 probability
      if (who_is_serving == 0) {
        probabilities <- c(my_win_prob_opponent_serving, 1- my_win_prob_opponent_serving)
        who_won <- sample(outcomes, 1, prob =  probabilities)
        #If my opponent wins, she gets a point, otherwise I become the server
        if (who_won == 0) {
          her_points <- her_points + 1  
        } else {
          who_is_serving <- 1
        }
        
      } else {
        #if I'm serving, I use 0.6
        probabilities <- c(my_win_prob_while_serving, 1- my_win_prob_while_serving)
        who_won <- sample(outcomes, 1, prob =  probabilities)
        #If I win, I add a point to my total, otherwise the other player serves
         if (who_won == 1) {
          my_points <- my_points + 1  
        } else {
          who_is_serving <- 0
        }
      }
      
    }
    
    if (my_points == 21) {
      return(1)
    }
    else {
      return(0) 
    }
}

Now, i’ll use a for loop to simulate the game 10,000 times and see how many times I win out of those 10,000 to determine the probability.

count <- c()
simulation_runs <- 10000

for (i in seq(simulation_runs)) {
  game <- raquetball()
  count <- c(count, game)
}

probability_to_win <- sum(count) / length(count)

The probability that you will win a game is: 0.8263