The “bad at math” tax

So you may have heard that the biggest lottery jackpot in the history of the world is up for grabs tonight. Currently the estimated jackpot is $640 million. That’s a lot of cheeseburgers, let me tell you. So, should you buy a ticket?

In general, no. If you’re looking at it strictly from a mathematical standpoint, comparing your odds of winning to the expected payout, the lottery is one of the worst gambling bets you can make. First of all, the “house” (in this case the government or various state governments for the big multi-state lotteries like Mega Millions and Powerball) usually takes 50% of all sales right off the top. That means if you buy a $1 ticket, only fifty cents goes towards the jackpot–the other fifty cents disappears into the state coffers. For some people, this is reason enough to never play the lottery.

But OK, say you’re perfectly happy paying an extra 50% tax to the government in this case. Hey, lottery fund money usually gets earmarked for stuff people generally like, such as schools or parks and recreation and such, to encourage them to play. So forget about the 50% tax you paid on the ticket, and concentrate on that jackpot, baby!

Well, it’s still almost never a good bet. The way you evaluate a bet is to compare the odds of winning to the expected payout. In the case of the Mega Millions game, the odds of winning the jackpot are a little bit better than 1 in 176,000,000. That means for every 176,000,000 times you play the lottery, you would expect to win one time.  Sure, in practice maybe you’d win twice and maybe you wouldn’t win at all, but mathematically your expectation is that you’d lose your bet 175,999,999 times and you’d win the jackpot once. So in order for the lottery to be a “good bet,” the expected jackpot has to be larger than the expected loss.

Well, look at that, in this case, it is! 650,000,000 is much bigger than 176,000,000–so if you could repeat this bet over and over again, you’d expect to lose your $1 bet 175,999,999 times (for a total loss of $175,999,999) and expect to win the jackpot once, for an expected win of $650,000,000.  Your expected win minus your expected loss gives you your expected value, or EV, which, superficially in this case, is $474,000,001!

Before you rush out and buy a bucket-load of tickets, however, let me remind you of a few things.  The first is the payout. No matter what, even if you’re the sole winner of the jackpot, you wont get $650 million right away. Most lotteries offer the chance to take the jackpot payout as either a lump sum or as an annuity. The catch is, the lump sum payment is half of the jackpot amount. So right away, you’re “only” getting $325 million. But still, with an expected win of $325 million, your EV for this bet would still be $149,000,001, so it’s still a good bet, right? Well, no, because no matter what you do, you’ll have to pay taxes on that money. Between federal, state, and local taxes you can probably expect nearly 40% of the jackpot to disappear right away. So suddenly your $325 million has shrunk to $195 million.

But lo, your EV is still positive, to the tune of nearly $19 million. But there’s still something we’re leaving out. You may not be the only winner. In cases where two or more people pick the same winning numbers, the jackpot gets divided. And in this case, even just one other winner brings your EV down into negative territory. And with so many people chasing this huge windfall, there’s a really good chance there will be multiple winners.

So what about the annuity? With that option, you get the whole jackpot, but you get it in equal payments over 26 years (for the Mega Millions jackpot, anyhow–different lotteries have different annuity lengths.) So for a $650 million jackpot, you’d get 26 yearly payments of $25 million each. For smaller jackpots the annuity payments can keep your tax liability in a smaller tax bracket, but in this case, without looking at the tax brackets, I’m betting twenty five million clams a year will still get you to the highest tax bracket.

At this point I’m going to have to beg off. I am not nearly good enough at math to figure out your EV on a bet that takes 26 years to pay off in full. For one thing, the value of money changes over time so that a dollar today is worth more than a dollar tomorrow (in general, because of inflation, the value of a dollar decreases over time, so that the buying power of that first $25 million payment will be much higher than the buying power of the 26th payment) and for another, smarter people than me have shown already that in general, unless you can expect an interest rate return of more than 7% or so, the annuity is a better value.

But forget about all that for a moment. As I’ve show, the jackpot has to at minimum be over $176 million before you can even start hoping for a +EV bet out of the lottery, and that doesn’t happen very often. So even if this week it’s possible that the Mega Millions has a positive expectation, in general it (and all lotteries) is a complete sucker bet.

You know what though? I’m not going to tell you not to play. It’s not any of my business if you want to spend money on sucker bets. I’d be a hypocrite if I told you not to play, anyhow. I like to play craps (and bet the hardways, even), I’ve played my share of penny slots, and you better believe I spent a buck on a Mega Millions ticket today. I figure I got a dollar’s worth of entertainment today just walking around flashing my ticket and saying “This is the winner right here, don’t even bother buying one!”

So yeah, treat it as entertainment and not as an investment and you can do what you want. But I do have a few other pieces of advice to share with you, starting with:

Buy just one ticket.

I can’t stress this one enough. I hear this all the time: “I bought ten tickets, now I have TEN TIMES THE CHANCES TO WIN!” Yes, that’s absolutely true, you increased your odds of winning tenfold! But your chance of winning in the first place is so minuscule that multiplying that number by ten simply doesn’t make it much bigger. Look at it this way. Even if you bought ten percent of all possible number combinations, you’d still have a 90% chance of losing, and you’d have spent $17.6 million trying. So don’t bother, just get one ticket.

Play your “lucky” numbers, or get a machine pick–it doesn’t matter.

Every combination of numbers has an equal chance of coming up. There really isn’t any such thing as lucky numbers of course, but if you want to play your birthday and your kid’s birthday and your anniversary date, go ahead. It simply doesn’t matter. 5 25 42 19 2 18 has the same probability as 1 2 3 4 5 6. Really.

Don’t play 1 2 3 4 5 6.

This one is based on hearsay, and I’m too lazy to do research to verify or disprove it, so take it with a grain of salt. But, although 1 2 3 4 5 6 is just as likely to come up as any other combination of numbers, I’m told that lots of people play combinations such as 1 2 3 4 5 6 or 2 4 6 8 10 or whatever, thinking that they’ll have a much better chance of winning the entire jackpot by themselves if their numbers do hit, because they think they’re the only one who has thought of playing these combinations that seem intuitively to be unlikely. But you know what? You’re not the only person who has thought of it, just like you’re far from the only person who thinks “password” is a great password. So in general I’d stay away from “obvious” number combinations.

The balls in the hopper have no memory.

With each drawing, each combination is just as likely as any other to come up. Full stop. This means that no numbers are “due.” Don’t bother spending time combing through years of statistics trying to find out which numbers haven’t been pulled from the hopper as much as statistics would indicate they should be. It doesn’t matter. Each drawing is an independent event, and by definition independent events have no bearing on the outcomes of other independent events. If 1 2 3 4 5 6 wins tonight, it is exactly as likely to win next week.

This one trips up lots of people. The most common objection goes something like this: If I flip a fair coin 20 times, what are the odds it will come up heads all 20 times? This is easy to calculate, you simply multiply the odds that it will come up heads on the first flip times the odds it will come up heads on the second flip, and so on for every flip to 20. So that is 0.5 times itself 20 times, which comes to 0.000000954, or just about one in a million. But, here’s the kicker. If you’ve already flipped a fair coin 19 times and it has come up heads all 19 times, what are the odds it will come up heads the 20th time? Simple. 1 in 2. You might think it is some insanely improbable event because the odds of getting 20 heads in a row are almost one in a million–but that doesn’t matter. On the 20th flip, as on every flip before or after that, the odds of it coming up heads are 1 in 2.

Look at it this way. Every time you flip a coin 20 times, you end up with a specific sequence of heads and tails that had only a one in a million probability of coming up. All tails is also 1 in a million.  Ten heads followed by ten tails is one in a million. Exactly H T T H H H T T H H H H T T T H T H T T is a 1 in a million shot. No matter what, after that 19th flip, you’re always just one head or one tail away from a specific 1 in a million run!

So don’t bother looking for streaks or hot numbers or cold numbers. Every combination is equally likely.

No, I’m not wrong because of “regression to the mean.”

Regression to the mean is the observation that the longer you flip a fair coin, the more likely you are to end up with 50% heads and 50% tails. This is true. What is not true, ever, is saying that “Because it has come up heads four times in a row, it is now more likely to come up tails, because of regression to the mean.” No. Don’t say that. It’s not true. Yes, it is true that over time you will get half heads and half tails. But that has no bearing on each individual flip–each flip is still 1 in 2, still just as likely to come up heads as tails. Similarly, saying that the number 24 hasn’t come up as many times as you’d expect over the last ten or fifty or five hundred lottery drawings, so it must be “due” to come up because of regression to the mean, is just wrong. Saying “regression to the mean” doesn’t make you right, trust me on this.

So there you go. Go ahead and play Mega Millions if you want.  It may even be a good bet this week. But you still shouldn’t expect to win.