2 min read

A simple probability problem

Amy Hogan, a stats and maths teacher who blogs at A Little Stats, posted the following quiz on twitter:

(Assuming fair dice) which has the highest probability:

1 six from 6 dice

2 sixes from 12 dice

3 sixes from 18 dice

The calculations aren’t too hard even by hand, and we have pbinom() available (if we remember to check \(<\) vs \(\le\) conditions). In that sense the question is easy, but I was looking for an intuitive argument. 

Obviously, the probability of exactly \(n\) sixes from \(6n\) dice is decreasing in \(n\), because the distribution is becoming less discrete. On the other hand, the probability of more than \(n\) sixes is increasing towards 1/2, since the distribution is becoming more symmetric. It isn’t obvious to me which one wins. 

Although I’d never encountered this before, it turns out to be a real classic. Isaac Newton answered it for Samuel Pepys, and got the brute-force calculations right, but then came up with an incorrect heuristic argument. Stephen Stigler has a paper, Joe Blitzstein pointed me to it before I wasted too much time. 

The neatest relevant fact is that the difference between the median and mean of a Binomial distribution is strictly less than 1, and so when the mean is an integer the two are equal. That implies the sequence \(P[\mathrm{Bin}(nk,1/k)\ge n]\) will tend to decrease with increasing \(n\) for any \(k\), but even that doesn’t quite prove the sequence is strictly monotone: we only know the probability is between \(0.5\) and \(0.5+P[\mathrm{Bin}(nk,1/k)=n]\). Also, there’s apparently no simple intuition behind the bound on the difference between mean and median. 

In the end, it turns out to be true that \(P[\mathrm{Bin}(nk,1/k)\ge n]\) is decreasing in \(n\) for any integer \(k\),  but (pretty obviously) \(\mathrm{Bin}(nk,p)\)  doesn’t have to be decreasing with \(n\) for general \(p\). Any valid intuition has to take advantage of \(p=1/k\). Stigler seems to think that’s an important barrier; I’m not convinced. Perhaps more off-putting,  any valid intuitive argument would probably have to make it obvious that the mean and median were equal when the mean is an integer.