## Lottery as an Investment

Lottery is a tax on people bad at math.

In this article I calculate how bad the lottery is as an investment, using Mega Millions as an example. To play the game, a player pays $1.00 and picks five numbers from 1 to 56 (white balls) and one additional number from 1 to 46 (the Mega Ball number, a yellow ball).

During the drawing, five white balls out of 56 are picked randomly, and, likewise, one yellow ball out of 46 is also picked independently at random. The winnings depend on how many numbers out of the ones that a player picks coincide with the numbers on the balls that have been drawn.

So what is your expected gain if you buy a ticket? We know that only half of the money goes to payouts. Can you conclude that your return is 50%?

The answer is no. The mathematical expectation of every game is different. It depends on the jackpot and the number of players. The more players, the bigger is the probability that the jackpot will be split.

Every Mega Millions playslip has odds printed on the back side. The odds of hitting the jackpot are 1 in 175,711,536. This number is easy to calculate: it is (56 choose 5) times 46.

How much is 175,711,536? Let’s try a comparison. The government estimates that in the US we have 1.3 deaths per 100 million vehicle miles. If you drive one mile to buy a ticket and one mile back, your probability to die is 2.6/100,000,000. The probability of dying in a car accident while you drive one mile to buy a lottery ticket is five times higher than the probability of winning the jackpot.

Suppose you buy 100 tickets twice a week. That is, you spend $10,000 a year. You will need to live for 1,000 years in order to make your chances of winning the jackpot be one out of 10. For all practical purposes, the chance of winning the jackpot are zero.

As the probability of winning the jackpot is zero, we do not need to include it in our estimate of the expected return. If you count all other payouts then you are likely to get back 18 cents for every dollar you invest. You are guaranteed to lose 82% of your money. If you spend $1000 a year on lottery tickets, on average you will lose $820 every year.

If you do not buy a lot of tickets your probability of a big win is close to zero. For example, the probability of winning $250,000 (that is guessing all white balls, and not guessing a yellow ball) by buying one ticket is about 1 in 4 million. The probability of winning $10,000 — the next largest win — is close to 1 in 700,000. If we say that you have no chance at these winnings anyway, then your expected return is even less: it is 10 cents per every dollar you invest.

You might ask what happens if we pool our money together. When a lot of tickets are bought then the probability of winning the jackpot stops being zero. I will write about this topic later. For now this is what I would like you to remember. From every dollar ticket:

- 50 cents goes to the state
- 32 cents towards the jackpot
- 18 cents to other winners

I am not at all trying to persuade you not to buy tickets. Lottery tickets have some entertainment value: they allow you to briefly dream about what you would do with those millions of dollars. But I am trying to persuade you not to buy lottery as an investment and not to put more hope into it than it deserves. If you treat lottery tickets as tickets to a movie that is played in your head, you will never buy more than one ticket at a time.

That is it. I advise you not to buy more than one ticket at a time. One ticket will allow you to dream about the expression on your sister’s face when she sees your new $5,000,000 mansion, but will not destroy your finances.

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## colorblind:

“for all practical purposes”

Playing the lottery is NEVER a practical purpose.

I remember back in college when I had to do laundry at the local laundromat. In those instances when I would forget my reading materials for the wait, I would head over to the convenience store next door and pick up a $2 crossword ticket. For that particular scratch-off, 67% goes back as prizes, about 30% are winners and the ticket, if played “correctly” (you scratch off letters to see how many “words” you can find in the corresponding “crossword” grid ), would take about 20 minutes to play – just about enough time for my laundry load to finish. If I won anything, I’d go back to the store, and use the winnings to purchase another ticket for the drying time. At worst, I was mildly entertained for $2. In a few instances, I won $20 – $50 and used it to buy flowers at the shop for my girlfriend [which was worth more than the winnings ]. For me personally, the entertainment value easily offset the expected loss of 68 cents.

31 December 2009, 6:44 pm## Eka:

I havent been able to digest the calculation of the expected payout since i have no understanding in the game..

31 December 2009, 8:42 pmOne thing for sure, i do agree with putting lottery as a mere entertainment but not an investment. Dont let lottery destroy our mood–that if we keep on failing..

## Pseudonym:

I’ve known people in jurisdictions where the lottery has no “profits” as such: all proceeds go to fund local government services. In such a location, partaking in the lottery could be seen to be entirely practical. It would be a voluntary tax for those who believe that schools are under-funded.

What’s interesting to me is that it’s NOT true that there is essentially no strategy for playing a lottery. Consider, for example, the degenerate case where you decide to buy 175,711,536 tickets in one draw. It would be incorrect to buy them all with the same combination of numbers. Buying one combination each would maximise your return.

Obviously, buying that many tickets is not practical. But if you buy fewer tickets, then the goal is not to improve your odds of winning, but reduce the probability of the “worst case”, where you don’t get your expected 18c return.

If you ignore the “mega ball” for simplicity, the problem boils down to the combinatorial covering design problem.

2 January 2010, 6:54 pm## sympath:

It’s not as simple as 50% going to prizes. That’s on average. In many large jackpot lottery games (like MegaMillions), the max payout starts very low, but if no one wins, the jackpot goes up. When there’s a collective losing streak, the payout can get large enough to exceed the odds, so that one could theoretically buy every combination and guarantee a prize larger than the investment. EXCEPT, when the prize gets large, more people play, and the chances of having to share the prize increase. The prize ends up being half the income from the entire run, but you don’t have to play at the beginning when the ‘investment’ is particularly bad. Thus I justify my occasional ticket purchase.

12 January 2010, 12:48 am## js:

Any expected value calculation, particularly for very large lotteries, should treat the value of a win as much larger than its actual dollar value – enormously so. In fact, it’s difficult to pin a value on the independence and opportunity that such a prize affords. So I think you can be sane and mathematically literate and still regard a lottery ticket purchase as a reasonable choice, not just based on entertainment value, as long as you can afford it without ruinous consequences.

13 January 2010, 11:29 am## Bill:

I agree with JS. If a lottery win would change my life drastically, while the financial loss from a losing ticket would barely be felt, it doesn’t much matter whether the payoff matches the actual mathematical odds.

As a kid, I had no conceptions of such things, and only saw the lottery as a fool’s game. My mother sent me to the store for a lottery ticket once. I had to choose 7 numbers from 01 to 80. To make my point, I selected the numbers 01 02 03 04 05 06 07. She got angry with me. “That’ll never come up.” I was not able to convince her that my picks had the exact same chances as the family birthdays or any other combination of numbers. My father’s view was more philosophical. He used to say “I only ever buy one ticket, because to win would take a miracle from God, and God only needs one ticket.”

Today, I play the scratch-off tickets. I like your image of a ticket to a movie that is played in your head. That’s exactly what it is for me. I buy one ticket at a time, and hold on to it for about a week. I leave it out on my desk, unscratched, and every time I look at it, it makes me smile. I might already be incredibly wealthy, after all.

19 January 2010, 11:22 am## Tanya Khovanova’s Math Blog » Blog Archive » The Expected Return on Lotteries:

[…] one of my previous pieces, I discussed returns on the Mega Millions lottery game, assuming that you buy a small number of tickets. In such a case winning the jackpot has zero […]

2 February 2010, 10:39 am## Powermillions:

Do not let math fool you. Someone will win, that is a fact.

13 July 2010, 9:03 pm