In case you haven't heard, the Mega Millions lottery reached an astonishing $648 million this week. There were two winning tickets, but

only one winner has come forward so far.

Here we'll look at the question of how much money you could expect to win from this lottery. In other words, for every dollar you spend playing the Mega Millions lottery, how much money could you expect to get back?

For reference, here's how some other investments stack up:

- If you invested $1.00 in the stock market a year ago, you would have $1.23 today. So for every dollar you invested, you would have made an additional 23 cents.
- If you play roulette at a casino, for every $1.00 you bet, you make, on average, $0.95 back. So for every dollar you bet, you would lose 5 cents.

So it looks like investing in the stock market is a pretty good way to spend a dollar, while playing roulette will lose you some money. How does playing the lottery stack up?

First, we need to know how likely it is for you to win different prizes. In each Mega Millions game you play, you select 5 numbers between 1 and 75, and then one addition number between 1 and 15. Here's what five games look like:

We could calculate the

**probability** of getting all 6 numbers right (using a branch of math called

**combinatorics**), but we'll skip that step for now. The Mega Millions lottery is very up-front with the probabilities of winning. Here's the chart they have on their

website, where the left-most column is how many of the five numbers you match (form 1 to 75), and the next column is whether you match that sixth number:

They advertise at the bottom that your chances of "winning any prize" are 1 in 14.7, or about 6.8%, which sounds pretty good. Unfortunately, just about all of that 6.8% is taken up by prizes of $5 or less.

Anyway, let's figure this out: for every $1 you spend on the Mega Millions, how much could you expect to win? In other words, we want to find the **expected value** of your cash winnings.

Here's a simpler example: suppose you roll a fair die (numbered 1-6), and you win a number of dollars equal to the number that comes up on a roll. So you have a 1 in 6 chance of getting $1, $2, $3, $4, $5, or $6. What's the average amount (or **expected value**) of money you'd make from this game? To find out, you can add up the probabilities of each event by the outcome of that event. In other words, you can expect to make:

which equals $3.50.

The expected value of the Mega Millions lottery without the jackpot is only about $0.18. But if we include a jackpot of $648 million, the calculation becomes a little more challenging, because that prize can be split if there's more than one winning ticket.

As far as we know, the Mega Millions lottery doesn't publicly announce how many tickets are sold, but it's probably in the many hundreds of millions for big jackpots like this one. As more tickets get sold, it's more likely that there are multiple winners. Let's say enough tickets were sold so that we can expect (there's that word again) about 2 winners.

With these numbers, you can expect to make $1.42 off of every dollar you put in the lottery. Those are better results than the stock market! Any time the lottery exceeds about $250,000,000, your expected winnings are greater than $1.00, so it *seems* like a good idea to play.

But not so fast. While you're definitely spending a full dollar to play, your winnings are taxable. The government may not tax you much when you win $50 for matching a few numbers, but you can bet they'll tax your jackpot prize (or the $1 million prize for matching 6 numbers). Assuming a 40% tax rate, for every $1.00 you spend, you'll now make only about $0.90, meaning you'll lose 10 cents. And you actually lose at lot more (closer to 40 cents) if you decide to take your winnings all at once. The 10-cent loss is only if you let the lottery make smaller payouts to you over the course of 20 years (and they don't account for inflation).

So to summarize, for every $1.00 you spent on Mega Millions, you could expect to lose about 40 cents. You're better off playing roulette.