Class activity solutions

Probability simulation: theater seating

Suppose there are exactly 100 seats in a movie theater. Further, suppose the movie theater has imposed a new ticketing system where each person’s ticket has an assigned seat number. Consider the case where the movie is sold-out (all 100 seats will be taken) and the first person in the theater lost their ticket and takes a random seat. Each subsequent person attempts to go to their seat. If their seat is open, they take it. Otherwise, to avoid confrontation, they choose a random seat.

1. Conduct a simulation to assess the probability that the last person into the theater will get their assigned seat.

Solution:

For loop approach:

set.seed(771)

n_people <- 100 # number of people in the theater
nsim <- 1000 # number of simulations to estimate probability
results <- rep(NA, nsim)
seats <- 1:n_people

for(i in 1:nsim){
  # vector to store which seats are taken
  # taken[i] is 0 when the seat is free
  taken <- rep(0, n_people)
  
  # first person randomly chooses a seat
  choice <- sample(seats, 1)
  taken[choice] <- 1
  
  # now go through everyone else (except the last person)
  for(j in 2:(n_people - 1)){
    
    # if the seat is free, take their seat. Otherwise,
    # randomly choose a seat from the ones available
    choice <- ifelse(taken[j] == 0, j, 
                     sample(seats[taken == 0], 1))
    taken[choice] <- 1
  }
  
  results[i] <- taken[n_people]
}

mean(results)

[1] 0.499

Notes:

• Make sure to use difference indices for the two for loops! Here I have used i and j
• In the inner for loop, I used 2:(n_people - 1) to do 2, 3, ..., 99. This is different from 2:n_people - 1, which would give us 1, 2, ..., 99.
• There are lots of other ways to code this question (e.g., you could use a counter to record the results, rather than a vector)
• It is crucial to treat the first person separately from the rest

Map approach: One run of the simulation is actually easier to do with the for loop, because the first person is treated differently than the others, and we need to update the taken vector. However, we can write a function to do one run of the simulation, then iterate it with map.

library(tidyverse)
set.seed(771)

n_people <- 100 # number of people in the theater
nsim <- 1000 # number of simulations to estimate probability

take_seats <- function(n){
  seats <- 1:n
  taken <- rep(0, n)
  
  # first person randomly chooses a seat
  choice <- sample(seats, 1)
  taken[choice] <- 1
  
  # now go through everyone else (except the last person)
  for(j in 2:(n - 1)){
    
    # if the seat is free, take their seat. Otherwise,
    # randomly choose a seat from the ones available
    choice <- ifelse(taken[j] == 0, j, 
                     sample(seats[taken == 0], 1))
    taken[choice] <- 1
  }
  
  taken[n_people]
}

results <- map_dbl(1:nsim, function(i) take_seats(n_people))

mean(results)

[1] 0.499