## Introduction

This guide details the expected profits, expected number of tickets sold, and likelihood of suffering a financial loss from each lottery jackpot level.

The main conclusions of this guide can be found in the "Bottom Line Up Front" section. Further detail can be found in subsequent sections. The math used to generate these conclusions can be found in the section at the end.

This guide is subject to change. Please comment with any questions, desired data, or corrections in order to help improve it.

## Bottom Line Up Front

The $5,000 jackpot is the worst possible lottery, in terms of both expected profit and probability of a loss. Thereafter, the larger the jackpot, the greater the expected profit and the lower the likelihood of a loss.

Statistically, the $1 k and $5 k jackpots will result in net losses. The $10 k jackpot offers a tiny but positive expected profit. Serious profits begin to accrue starting at the $50 k jackpot, which offers $3.5 k in expected profit. All lotteries are risky; even the $1,000 k jackpot has a 40% chance of losing money on any given run, even if it will likely gain profit over many repetitions.

The table below details the findings. A table legend can be found beneath for clarity on what each column represents.

Jackpot ($, thousands)Ticket Cost ($)Win Chance (%)Expected Tickets Until WinExpected Profit ($)Tickets to Break EvenLikelihood of Loss (%)$1 k$352.50%27.3-$41.782951.5%$5 k$1251.80%38.1-$229.934051.6%$10 k$1751.20%57.4$47.625849.8%$50 k$5500.71%97.2$3,503.669147.7%$100 k$8000.47%147.1$17,705.1212544.5%$250 k$1,5000.33%209.7$64,546.7516742.4%$1,000 k$4,0000.20%346.2$384,907.6025039.4%

Jackpot: The size of the jackpot being offered. Values are measured in thousands of dollars, as denoted by adding "k" to the end. (E.g., Five thousand dollars would be written as "$5 k")

Ticket Cost: How much customers pay for each ticket.

Win Chance: How likely each ticket is to win.

Expected Tickets Until Win: Over an infinite number of plays, how many tickets will be sold on average before a winning ticket is sold and the lottery ends.

Expected Profit: Over an infinite number of runs, how much profit (revenue minus costs) will be made on average from the lottery.

Tickets to Break Even: How many tickets must be sold for the lottery to begin generating profits.

Likelihood of Loss: How likely it is that a winning ticket will be sold early enough for the lottery to lose money.

## Odds Of Win On Or By Nth Ticket

This section presents the likelihood that a lottery will have a winner on the Nth ticket sold, and by the Nth ticket sold. The statistics for a win on the Nth ticket sold are not very informative; it starts high and decreases asymptotically toward zero for all jackpots. it is computed in order to calculate the cumulative chance of a win, described as a winner by the Nth ticket sold. It also allows computation of the expected profit distribution, discussed later.

The probability of a win is shown in the two figures below. They are broken apart into small lottery jackpots, under $100,000, and large ones which are $100,000 or more.

Vertical dotted lines, denoted "$X k BE", denote the number of tickets required to break even and begin turning a profit.

Because the probabilities are additive, it is helpful to examine the cumulative probability distribution function (CDF). These are displayed below.

As above, the dotted line denotes the breakeven point. Note that for the $1 k and $5 k jackpots, the odds of a win by the breakeven ticket are slightly above 50%, meaning it is more likely someone will win before you have made your original money back. For the $50 k and above jackpots, they reach breakeven before there is a 50% chance of a win.

These values can be used to determine the likelihood of a loss. Alternatively, it may be calculated directly as described in the Math section.

## Expected Profit Of A Win On Nth Ticket

It's always possible, however unlikely, that the first ticket in the $1,000 k jackpot will be a winner. It's also possible that the $1 k jackpot will go on for hundreds of tickets. However, these outcomes are unlikely, and over the long run few losses or profits will be derived from these outcomes. More useful would be a way to gauge the relative risk of each lottery to figure out how much money one is likely to gain or lose for a range of possibilities.

This is done by multiplying the likelihood of a win by the Nth ticket by the profit or loss generated by said win. For example, losing $1,000 k on the first ticket would result in a loss of $996 k to the player, but only happens 0.20% of the time, so the weighted value of that loss is -$1,992. In contrast, the odds of winning on exactly the 500th ticket are 0.7%, and would generate $500 k in profit for the player, so its weighted value is $736.50

The probability values above were combined with the profit and loss for each ticket level. The following charts show the distribution of expected profit. To interpret it, consider the relative areas between the curve and the X-axis. A larger area indicates a higher probable profit or, if the area is below the axis, a higher probable loss.

## The Math Behind The Tables

This section details the math used to generate the projections. It is presented so others may make their own spreadsheets, point out errors, or trust in the guide.

The BasicsThe fundamental equation governing this guide is given by X = q^n where:

X = The probability of n failures (tickets that do not win) in a row

q = 1 - win chance = the probability of any given ticket NOT winning

n = The number of tickets sold so far

Finding the Expected Number of Tickets Until WinIf we wish to find the point at which some portion of all lotteries should have had a winner, we solve for n.

X = q^n

ln(X) = ln(q^n) = n ln(q)

n = ln(X)/ln(q)

So, to find where half of all lotteries have won, we set X=0.5 and use the q given by the lottery jackpot level. Note that we can also find the range of outcomes by using other values for X. However, unless you choose very small deviations (45th versus 55th percentile) you are unlikely to find useful data in this case.

Remaining CalculationsThe expected profit is straightforward to calculate. Using the above n, multiply by the ticket cost, and subtract the jackpot.

Similarly, the tickets required to break even can be found by dividing the jackpot by the ticket cost and rounding up.

Probability of Winning on Nth TicketThe probability of winning on the Nth ticket sold is given by the probability of that win, p, multiplied by the probability of each ticket that came before being a loss, q^(n-1). Therefore, the probability of winning on the Nth ticket W(N) = q^(n-1) * p.

Likelihood of LossUsing the above required break even level (without rounding), one can calculate how likely it is that any given lottery will conclude before it begins to generate profit. Let B be the number of tickets required to break even. Using our fundamental equation X = q^n, We know that the probability of breaking even is q^B. The probability of not breaking even and getting a loss is therefore 1 - q^B.

Source: https://steamcommunity.com/sharedfiles/filedetails/?id=3332500124

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