I’m not an especially big fan of the lottery — I consider it to be a tax on those who can least afford to pay the tax, but do so because they think it gives them a chance at a better life.

Nonetheless, this week’s record \$500 million jackpot makes for an interesting mathematics exercise as those of us who don’t normally waste money on lottery tickets ask ourselves, “is it worth it to buy one”?

### Odds vs Expected Return

The Mega Millions website very clearly gives the odds of winning each prize. If you buy a single \$1 ticket, you have a 1/40 chance of winning any prize at all and a 1/175 million chance of winning the jackpot.

But a different and, perhaps, more meaningful question is the expected return on your “investment” of a \$1 ticket. In computing the expected return, we weight the odds according to the prizes.

Note that “expected return” doesn’t actually mean that you “expect” to receive that particular dollar amount. For instance, if I were to offer you the option to flip a coin and if it comes up heads, I will give you \$5 and if it comes up tails, I will give you \$10, your “expected return” is \$7.50, even though there is no chance whatsoever that you will receive that amount. Nor is “expected return” the same thing as “most likely return” — your most likely return on buying a lottery ticket is, of course, that you have just made a \$1 donation to the government and will get nothing.

So, consider the eight lower prizes (we’ll look at the jackpot separately):

Match Match Prize Odds Expected Return
5 0 \$250K 1/3,904,701 \$0.064
4 1 \$10K 1/689,065 \$0.015
4 0 \$150 1/15,313 \$0.010
3 1 \$150 1/13,781 \$0.011
3 0 \$7 1/306 \$0.023
2 1 \$10 1/844 \$0.012
1 1 \$3 1/141 \$0.021
0 1 \$2 1/74.8 \$0.027
cumulative 1/40 \$0.182
(Amounts do not add up exactly due to rounding)

So not counting the jackpot, your expected return or average return on a one dollar Mega Millions lottery ticket is 18.2 cents.

### Computing the Jackpot Expected Return

You might expect that since the jackpot for Friday’s drawing is \$500 million that the expected return is 500,000,000/175,711,536, or, \$2.85 on a \$1 ticket. Unfortunately, this is not the case. There are three considerations that we need to make:

1. The cash option vs. the advertised jackpot
2. Taxes
3. The possibility of multiple winners

#### Cash Option vs. Advertised Jackpot

The first consideration is the most obvious – \$500 million is the projected future value of investing \$359 million in an annuity made up of ultra-safe government bonds that, as long as the U.S. doesn’t default on its debt, are pretty well guaranteed to pay out. Despite last year’s hysterics, the United States has never been in any serious jeopardy of defaulting on its debt (even if we don’t raise the debt ceiling at some point, tax revenue more than covers debt payments) and so as long as the country exists, this annuity is safe.

Nonetheless, it doesn’t make sense to use the annuity value in ROI (return on investment) calculations any more than it would make sense for your employer to quote your salary in terms of what it would be worth in 20 years if you invest it in a particular stock. So the “real” jackpot is \$359 million.

So we might expect that the expecting return is \$359,000,000/175,711,536, or, \$2.04 on a \$1 ticket — still a bargain. Unfortunately, we next need to consider taxes.

#### Taxes on Lottery Winnings

You will often read, even from supposedly reliable news outlets, that there is some 25% “gambling tax” and so you can expect 25% of your \$359 million to go to Uncle Sam. That is not correct. The source of this confusion is that the IRS requires a 25% withholding on gambling proceeds over a certain amount, but the actual tax that you owe is not going to be 25%. Gambling proceeds are treated as ordinary income and if you hit the jackpot, that will put you into the top bracket, which is currently 35%. All income over \$388,350 is taxed at 35% and so if you make \$359 million, your tax is approximately \$125.6 million.

Similarly, in Virginia, even though only 4% is withheld, gambling proceeds are ordinary income and Virginia’s top marginal rate is 5.75%. So if you were to win Mega Millions in Virginia, you can expect to pay a total of 40.75% in taxes, or, \$146 million.

This tax, of course, may be reduced if you donate a portion of your winnings or invest them in a tax-deferred asset. But if you simply picked up a check, you would pay \$146 million in taxes. So we’re going to consider \$213 million as our “after tax” amount.

### The possibility of multiple winners

Finally, we need to consider the very real likelihood that you will not be the only winner.

Let’s look at a simple case first. Suppose that you and a friend each pick try to guess a randomly generated number from one to ten. I think we would agree that each of you individually has a 1/10 chance of guessing right. But there is a 1/100 chance that you are BOTH going to be right. So, the odds would be:

• 1/100 = both right
• 9/100 = you and you only are right (1/10 – 1/100)
• 81/100 = you are both wrong

So it’s only slightly more complicated when we look at the lottery. Suppose that you and your friend both pick one random ticket:

• 1/(175 million)2 = both win
• 1/(175 million) – 1/(175 million)2 = you and you only win
• 1/(175 million) – 1/(175 million)2 = your friend and only your friend wins
• (1/(175 million) – 1/(175 million)2)2 = you both lose

Now, if only two people in the whole universe played the lottery, it would be essentially impossible for them both to win. But when hundreds of millions play, you are actually more likely to have multiple winners than to just have a single winner.

In calculating the odds of multiple winners vs a single winner, we use something called a Poisson distribution.

This table shows the odds of having a particular number of winners, given the number of people who play the lottery. So if 150 million tickets are sold, there is a 42.58% chance of having no winners, a 36.35% chance of having exactly one winner, etc.

Tickets sold No winners 1 2 3 4 5
10 mil 94.47% 5.38% 0.15% 0.00% 0.00% 0.00%
20 mil 89.24% 10.16% 0.58% 0.02% 0.00% 0.00%
30 mil 84.30% 14.39% 1.23% 0.07% 0.00% 0.00%
40 mil 79.64% 18.13% 2.06% 0.16% 0.01% 0.00%
50 mil 75.23% 21.41% 3.05% 0.29% 0.02% 0.00%
60 mil 71.07% 24.27% 4.14% 0.47% 0.04% 0.00%
70 mil 67.14% 26.75% 5.33% 0.71% 0.07% 0.01%
80 mil 63.43% 28.88% 6.57% 1.00% 0.11% 0.01%
90 mil 59.92% 30.69% 7.86% 1.34% 0.17% 0.02%
100 mil 56.60% 32.21% 9.17% 1.74% 0.25% 0.03%
120 mil 50.51% 34.50% 11.78% 2.68% 0.46% 0.06%
150 mil 42.58% 36.35% 15.52% 4.42% 0.94% 0.16%
175 mil 36.94% 36.79% 18.32% 6.08% 1.51% 0.30%
200 mil 32.04% 36.47% 20.75% 7.87% 2.24% 0.51%
250 mil 24.10% 34.30% 24.40% 11.57% 4.12% 1.17%
300 mil 18.13% 30.96% 26.43% 15.04% 6.42% 2.19%
350 mil 13.64% 27.18% 27.07% 17.97% 8.95% 3.57%
400 mil 10.26% 23.37% 26.60% 20.18% 11.49% 5.23%

But now we have to ask, how many people are playing? As is no doubt obvious, the number of players varies from week to week depending on the jackpot size. Smaller jackpots yield a small number of players and large jackpots yield a large number of players.

The lottery doesn’t tell us exactly how many tickets were sold, but we can estimate the number of players based on how many of the lower-level prizes were handed out. On the most recent drawing, 2.53 million tickets won the \$2 prize for matching only the yellow ball, which has approximately 1/74.8 odds, so we can extrapolate that there were around 190 million tickets sold. 1.35 million tickets won the \$3 prize (1/141 odds), which also comes out to around 190 million.

So for the recent drawings going back through the beginning of March, here are the numbers of tickets purchased we have seen:

Jackpot (millions) Tickets (millions)
363 190
290 111
241 75
200 76
171 48
148 49
127 37
108 41

Now, the way the estimated jackpot is computed is that it is based on the number of tickets that the lottery expects to be purchased (as opposed to basing it on the previous drawing’s sales and then using this drawing’s sales for next week). So extrapolating previous jackpots vs ticket sales, we can guess that with a \$500 million advertised jackpot, Mega Millions is expecting around 350 million tickets to be sold.

### So what is my expected return?

Plugging our ticket sales guestimate into our table above, we can see that there is a 27.18% chance of exactly one winner, a 27.07% chance of exactly two winners, etc.

So presupposing that you do possess the winning ticket (an event that has a 1/175711536 chance of happening), your expected prize, after taxes, is:

\$213 million / * 27.18% +
\$213 million / 2 * 27.07% +
\$213 million / 3 * 17.97% +
\$213 million / 4 * 8.95% +
\$213 million / 5 * 3.57% +
\$213 million / 6 * 1.18% +
\$213 million / 7 * 0.34% +
\$213 million / 8 * 0.08% +
\$213 million / 9 * 0.02%

That adds up to \$106.3 million. But we’re double-counting the 13.64% chance that nobody wins, so actually, it’s \$123.1 million. (Remember, we’re presupposing you have the winning ticket … so we’re not interested in the possibility that nobody wins.)

So your expected return for the jackpot is 70.1 cents. Added to the cumulative expected return on all other prizes of 18.2 cents, for your “investment” of a \$1 lottery ticket, your expected return is 88.3 cents for this Friday’s drawing.

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