Wikipedia:Reference desk/Archives/Mathematics/2018 August 2

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August 2

Voting result distribution

There is a country with 10000 people and there is an election. 6734 people showed up. Is the last number, 4 in this example, completely random or is some number more likely? — Preceding unsigned comment added by 193.64.221.25 (talk • contribs) 09:07, 2 August 2018 (UTC)

If by "completely random", you mean that it would follow a discrete uniform distribution, then yes, the last digit is completely random. Iffy★Chat -- 09:32, 2 August 2018 (UTC)

Well, it would be random and uniformly distributed for all practical purposes, assuming there isn't some unusual circumstance. If there are 10000 people each with a p=.67 chance of voting, and whether a person votes is independent of the others, then the number of people voting follows a binomial distribution, from which you could, in theory, work out the probability of the last digit being 4. For p=.67 this is presumably very close to .1, but in terms of p it's a polynomial which takes on value 0 at p=0 and p=1, so not a constant .1. It might be that there are 6733 people who always vote no matter what, and 1 person who has a .5 chance of voting, in which case the probability of the last digit being 4 is .5. (This is why you need the unusual circumstances clause.) In reality elections are more complicated: some people vote more often than others, some people may vote as a couple so one votes only if the other does, and some people may watch the early results and only vote if it looks close. There is probably still enough mixing to presume approximately probability .1 for each last digit, but if the population was only 20 then maybe not. See Numbers game#Gameplay for a similar idea put to 'practical' use. --RDBury (talk) 13:05, 2 August 2018 (UTC)

PS. It's actually possible to compute an upper bound for the difference between the uniform distribution and the distribution of the last digit of a variable with the binomial distribution. In general (assuming my algebra is correct) if X is a random variable with binomial distribution B(n, p), and X_k is the last digit of X base k, then the difference between P(X_k = d) and 1/k is less than

$(1-4p(1-p)\sin ^{2}{\tfrac {\pi }{k}})^{n}.$

In the case k=10, p=.67, and n=10000 this works out to 6.4e-384, which is negligible but not actually 0. --RDBury (talk) 02:24, 3 August 2018 (UTC)

Not sure whether Benford's Law is applicable here but it's an interesting article. manya (talk) 04:36, 10 August 2018 (UTC)