# Money for (almost) Nothing

Five Hundred Mega Dollars, to be precise. (Image from Wikipedia)

I am not typically interested in lotteries. They seem silly and I am seriously beginning to question their usefulness in bringing about a good harvest. But this morning I read in the news that the Mega Millions lottery currently has a world record jackpot up for grabs. In fact, the jackpot is so big…

Tonight Show Audience: *HOW BIG IS IT?*

**

It is *so big* that I decided to do a little bit of analysis on the
expected returns. Zing!

## Some Background

First, a little background. The Mega Millions lottery is an aptly named
lottery in which numbered ping pong balls are pulled from a giant
rotating tub of randomization. Five of these are drawn from one tub of
56 balls, with no replacement. The sixth ball (the so-called “Mega
Ball”) is drawn from a *separate*tub of 46 balls. To play, one picks 5
different numbers (1-56) for the regular draws and one number (1-46) for
the Mega Ball. The first five can match in any order, but the last ball
has to match with the Mega Ball. Prizes are given out based on how many
numbers you match. Stolen from the Mega Millions website, the prizes and
odds are given in the table below. The current jackpot is listed at

Don’t worry about the asterisk. It just says CA is lame. (Source: Mega Millions)

Hot diggity daffodil, we’re ready to get going!

## Expected Winnings

Alright, so it costs *feels* like it is worth it to play (what’s the harm?).
But we can do better than feelings, we have… MATH!

Since we have an exhaustive list of outcomes and their probabilities (which is just the inverse of the big number in the “chances” column), we can calculate the expectation value for our winnings. The expectation value is just the sum over all the possible prize values times the probability of winning that prize. In other words,

where we denote our expected winnings in angled brackets.

In essence, this value represents the average prize you would win if you played this lottery over and over and over again (or played all the combinations of numbers).

Setting the jackpot to $500 million, we can now compute the expected winnings as

A few flicks of the abacus later, we find that the expectation value of our prize is

which means that after we subtract the dollar we paid for the ticket, our expected return is $2.02.

But what if we had chosen to take our winnings as a lump sum of

which results in a $1.22 gain when we subtract the dollar we paid for the ticket.

At least in a statistical sense for this particular jackpot, one is better off playing than not playing. But are we forgetting anything?

## The Taxman

If you win a

After applying federal and state taxes to the prizes above $5000, we now have an expected winnings of

which gives an expected net win (minus the

We’re still in the black, but it’s slowly slipping away. Is there anything else we need to factor in? Well, yes. For one thing, winning the jackpot qualifies us for the top tax bracket, so most of the winnings would be taxed at the top marginal tax rate of 35%. Welcome to the 1%, kids! [1].

Changing the federal tax rate on the jackpot from 25% to 35% and
recalculating, we find net expected winnings of

## Is it always like this?

One thing to keep in mind as we make these estimates is that this is a
*historically large* jackpot. So even though it may be favorable to play
this time, this will not always be the case. In fact, we can find the
minimum jackpot value for which this is the case.

The condition in which our expected return is a gain (rather than a loss) is

For simplicity, let’s ignore the top marginal tax rate and just factor in the 25% withholding and the 6.8% state tax. Solving for the minimum jackpot using the expression for we found in the last section, we see that

Technically, this would have to be the amount actually awarded by the
payment method of your choice. The *stated* jackpot is always the
annuity method (because it looks higher). The lump sum offering is *at
most* about 70% of the stated jackpot. So if you want to take the lump
sum offering the *stated* jackpot will need to be

In fact, these values are likely a bit low, since we have not included the increase to the marginal tax rate, nor have we included other effects like having to split a prize (which seems to happen a lot) or inflation effects if you take the prize in yearly installments.

In any case, a quick look through the jackpot history shows that these threshold values are only met occasionally. An eyeball estimate puts about one jackpot per year that exceeds the (absolute) minimum $217 million threshold.

## So am I going to win?

No. No, you will not. BUT if you played record setting lotteries
hundreds of millions of times, you might see decent (\~10%) returns.
Although, it may just be easiest to, you know, *invest* that money.

#### Only One Useless Footnote

[1] Although, to be fair, the top marginal tax rate is currently at historical lows. It could always be worse… [back]

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