Dice Rolls

In a Straussian attempt to prove personality is genetic, my little brother texted me the following math question:

So imagine we have a die of n sides. We roll the die until it rolls a 1, the number of times it is rolled is the output. But, after each roll, we give the die one less face. What does the distribution of outcomes look like?

When I come across problems like this, I like to answer the question intuitively before solving for the actual answer. My brother and I both guessed what the distribution looked like. We both thought there’d be a very low chance of n or 1 being returned, and a higher chance of a number in the middle being returned. I thought the mode of the distribution would be lower than him.

Perhaps because probability wasn’t beneficial in the anscestral environment, we were both wrong. The distribution is actually perfectly uniform. Elementary math will show that the probability of rolling the first few numbers is exactly the same:

\frac{1}{n} , \frac{n-1}{n} \frac{1}{n-1} , \frac{n-1}{n} \frac{n-2}{n-1} \frac{1}{n-2}

I tested this empirically and produced the following histogram.


Here’s the R Code that generated that:

##Run Parameters
sides <- 100
runs <- 100000

Simulation <- as.data.frame(matrix(data=0,nrow=sides,ncol=runs))
for (n in 1:sides){
Simulation[n,] <- floor(runif(runs,1,sides-n+2))

Simulation <- Simulation==1

Outcomes <- vector(mode="integer",length=runs)

for (n in 1:runs){
Outcomes[n] <- which.max(Simulation[,n])