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What kind of probability distribution is best suited to describe lotteries?

Wikipedia provides extensive list of probability distributions (another list of them - see at bottom). But reading articles concerning applicability of each of those distributions I never see any direct hint that it can be used for lotteries.

So far I make the following conclusions:

  • It shall be something in "Discrete univariate with finite support" group, that is one of: Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

  • Binomial distribution is best suited to describe lotteries

I want to undestand what I can model in lotteries with probability distributions. For example, I can model how a real lottery deviates from theoretical probability (for example, concerning expected vs real frequency of appearing of number $7$ in in some lottery).

So far I think that normal (Gauss) distribution cannot be applied to lotteries and rule of $3$ sigmas also cannot be applied to lotteries. Am I right?

Also, Discrete uniform distribution seems to be a nice fit.

triplodrindl
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For your thoughts, in this Wikipedia page about binominal distribution, it does refer to the expression $\frac{n!}{k!(n-k)!}$, related to the lottery.

This is because in lottery, the best way to use is the formula to calculate the number of ways to choose $k$ distinct objects from $n$ objects, the order doesn't matter.

From Brilliant:

Consider the following example: Lisa has $12$ different ornaments and she wants to give $5$ ornaments to her mom as a birthday gift (the order of the gifts does not matter). How many ways can she do this?

We can think of Lisa giving her mom a first ornament, a second ornament, a third ornament, etc. This can be done in $\frac{12!}{7!}$ ways. However, Lisa’s mom is receiving all five ornaments at once, so the order Lisa decides on the ornaments does not matter. There are reorderings of the chosen ornaments, implying the total number of ways for Lisa to give her mom an unordered set of ornaments is $\frac{12!}{7!5!}$.

Notice that in the answer, the factorials in the denominator sum to the value in the numerator. This is not a coincidence. In general, the number of ways to pick $k$ unordered elements from an $n$ element set is $\frac{n!}{k!(n-k)!}$. This is a binomial coefficient.

There is also another Wikipedia article about this, and I think the best probability to apply is Probability: sampling a random combination at the end of the page:

There are various algorithms to pick out a random combination from a given set or list. Rejection sampling is extremely slow for large sample sizes. One way to select a $k$-combination efficiently from a population of size $n$ is to iterate across each element of the population, and at each step pick that element with a dynamically changing probability of $\frac{k-\text{#samples chosen}}{n-\text{#samples visited}}$ (see reservoir sampling).

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user061703
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  • Thank you for very nice links and explanations about combinations! I agree, but still I would like to know what probability distribution to use speaking about lotteries. – triplodrindl Apr 20 '18 at 10:40
  • If you take multiple independent measurements, on the macro scale you tend to get Gauss-distribution, independently of what each individual measurement's distribution is, as described in the central limit theorem. I have tried searching for extensive papers on the subject for fun but I didn't really find good ones on it. Neither I know if hypergeometric or normal distribution describes lottery in general better, based on pure definition, hypergeometric is more accurate. Maybe you could research this? – Balázs Börcsök Jun 13 '21 at 14:20
  • Also I would like to add that hypergeometric describes k successes from n draws without replacement which may not be a good measurement of saying, for instance "Which numbers may I need to pick, to maximase my chance of winning?". – Balázs Börcsök Jun 13 '21 at 14:24