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We study the mean earnings of a lottery winner as a function of the number

The game of lottery is both popular and simple. Focusing on the essentials, and leaving aside additional features of secondary importance, which vary across different lottery implementations, the rules of the game are as follows: each player submits to the lottery organizers a ticket consisting of

For a given success probability

An additional complication rollover presents that it is not only that the number of participants in the various draws can vary, but also that this variation can potentially exhibit strong correlation, specifically be strictly increasing: as a rule, larger money prizes motivate more extensive participation [

Needless to say, actual lottery implementations are actually more complicated, incorporating a plethora of secondary features. For example, lower winning levels may be introduced, corresponding to partial matches between the tickets and the winning set, and the prize money they win may either be smaller fractions of the total sum or simply a fixed sum. Alternatively, an extra “bonus number” may be introduced, acting, in a sense, like a “second chance”: players are allowed to submit a guess for this number, in addition to their guess for the winning set, while the selection of the winning set is followed by the (also uniformly random amongst the remaining integers) bonus number. If a player's choice has an overlap of

To the best of our knowledge, the issue of winners' mean earnings has not been considered in the lottery literature before. It is a problem of mathematical interest, relating lottery to the study of inverse moments of the positive binomial distribution (as will be seen below), and, for this reason, this article can be considered falling into the same group of publications by the present author [

One final point that needs to be addressed is our choice to assume throughout this article that players choose their numbers independently of each other, namely, that the probability distribution describing the choice of an

Consider the lottery game as described in the Introduction: the probability distribution for the number of winners is the binomial

This expression is essentially the inverse first moment of the positive binomial distribution

The problem of the inverse moments of the positive binomial distribution was first considered by Stephan in 1946 [

We now proceed to derive a more convenient formula for (

We can transform (

We now approximate the sum in (

The error of this approximation is readily given by the lowest-order Newton-Cotes numerical integration formula known as the midpoint rule [

The approximation (

Comparison of

Figure

For fixed

To begin with, note that

Putting (

Alternatively, the eventual positivity of

For fixed

In particular, the maximal mean earnings possible are about 29.5% above the asymptotic value, and about 31% of the total prize money.

Considering

What about maximizing the relative mean earnings

As a brief numerical example, consider the Greek lottery, a

Before we conclude this section, let us investigate the variance of a winner's earnings. Our goal is to obtain a similar formula as the one presented in Theorem

It follows that the variance is

Asymptotics can be found as in Theorem

The generalization is clear. For any

The relation to the inverse moments follows from the generalization of (

When rollover is introduced, a sequence of lottery draws is played till a winner is found, at which point the total prize money (consisting of the prize money of the current draw plus the

We denote the mean earnings under rollover by

It is, in fact, possible to determine the probability distribution of

It would be, of course, far more interesting to enrich the model by allowing

In general, (

Assuming that the number of tickets submitted remains constant throughout rollover draws, the asymptotic expression for

In particular, the maximal mean earnings possible are about 32% above the asymptotic value, and about 35% of the (first draw) total prize money.

Under the stated assumptions, (

We now note that the asymptotic behavior of

Continuing the numerical example for the Greek lottery given right below Theorem

We may attempt, as in Theorem

Figure

(a)

We investigated the mean earnings of a lottery winner as a function of the number

The author would like to thank the two anonymous referees for their detailed and insightful comments, which greatly improved this paper; in particular, for bringing to the author's attention that the problem studied here is related to the classical problem of determining the inverse moments of the binomial distribution, and for providing an extensive list of additional references.