Problem statement

Two teams play the “coin flipping game”. There is a small team that has 3 players, and a large team that has 20 players.

Here are the rules:

1. Each player flips a coin

2. We count the total number of Heads that a team gets

3. Divide the total number of Heads by the team size, to get the proportion of Heads

4. The team with the higher proportion of Heads wins the game

The question we are investigating is: is the smaller team at an advantage or a disadvantage in this game?

We know, of course, that smaller samples give more highly variable results. See Kahneman and Tversky’s example about babies’ gender proportions, for example.

However, in this case, I’m not sure that this higher variability will be either beneficial or detrimental for the smaller team.

Let’s simulate to find out!

Simulation setup

We’ll define a function to return a vector with the proportion of Heads for each iteration of the game.

# fn definition
head_prop_sims <- function(n_sims,
                           team_size){
    
    # n_sims: how many simulations are we running? 
    # team_size: this is the parameter n for the binomial random variable generator
    
    # example: 
    # for the first simulated result, the proportion of Heads 
    # will be a binomial r.v. with parameters n = team_size, p = 0.5 
    
    # same for the second simulated result, and so on. 
    
    # Vector of results:  
    return(rbinom(n_sims, 
                  team_size, 
                  prob = 0.5)/team_size)
}

# fn test 
# head_prop_sims(10, 2)

Analysis

Using our function with two different values for team_size, we get simulated Heads proportions for both the small and large teams.

n_iterations <- 100000
small_team_size <- 3
large_team_size <- 20

small_team_proportions <- head_prop_sims(n_iterations, small_team_size)
large_team_proportions <- head_prop_sims(n_iterations, large_team_size)

Next, we take the difference to see who wins each iteration.

small_team_advantage <- small_team_proportions - large_team_proportions

Finally, let’s plot the difference. If the difference is centred around zero, then team size has no impact on win probability.

data.frame(small_team_advantage = small_team_advantage) %>% 
    ggplot(aes(x = small_team_advantage)) + 
    geom_histogram() + 
    labs(title = "Distribution of the difference in proportion of Heads", 
         subtitle = sprintf("Each data point is the difference between the proportion of Heads \nthat the small team got and the proportion of Heads that the large team got \n\nBased on %i iterations", 
                            n_iterations))

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Conclusion

It doesn’t look like team size has an impact on win probability.