Playing the Lottery, part 2

In my previous post, I talked about a calculation called “expected value”, which helps measure just how fair or unfair a given game is. I also talked about “the gambler’s downfall”, which basically means that the player is much more likely to run out of money before the house does. In this post, I’ll talk about five ways the state lottery tries to trick you into thinking that the game is better than it really is.

#1 The prize might be divided among several winners. They want you to think about the size of the jackpot and ignore the fact that several winners may end up splitting the jackpot. A $72 million jackpot sounds bigger than a $24 million jackpot, but that’s just an illusion. The $72 million jackpot is much more likely to be split three ways, so each gets $24 million.

#2 They lie about the value of the jackpot. I’m not talking about taxes; that’s a whole other subject. Imagine a game where, if I win, you have to pay me right now, but if you win, I take 30 years to pay you. How fair does that sound? When they tell you that the prize is $24 million, that’s a deception. The truth is that they are essentially offering you a gift certificate which is only worth $14 million. You can trade it for $14 million in cash, or you can trade it for an annuity that pays $800K per year for the next 30 years. The problem here is the difference between Present Value and Future Value. $24 million is the Future Value, spread out over 30 years. But I don’t care what it will be worth in the future. What matters is what it’s worth right now. The Present Value is only $14 million, not 24. There exact ratio of Present Value to Future Value depends on interest rates, but right now PV is roughly 60% of FV over 30 years.

#3 They use huge numbers in order to confuse people. Most people can understand small numbers like $50 and $1,400 but they have trouble understanding just how big is a million, or a billion. The lottery takes advantage of this by offering what seems like a large prize and burying in the fine print the fact that the odds against you are even more astronomical. Sometimes it’s 14 million to 1 against, sometimes it’s 292 million to 1 against. Your brain sees both those numbers as just “really big”, even though the second one is twenty times higher.

#4 They make the game complicated. This has the double whammy of making it more fun (because it feels like you have some control) yet it also makes it harder to understand the odds. Even if you’re one of the rare people who learned Pascal’s formula for expected value, they are betting you won’t be able to apply it to such a complicated game. It has been said that the lottery is a tax on people who are bad at math. The truth is that even people who are relatively good at math have trouble understanding the lottery. Luckily, you have me to help you.

#5 The exact parameters of the game aren’t known until after it’s over. In order to figure out how much you might win, you need to know how many tickets will be sold. But that’s not known at the time you buy your ticket. And they are constantly adjusting their estimate of what the jackpot will be. In fact, the size of the jackpot also depends on how many tickets get sold.

Let’s try to estimate what the expected value of the lottery really is. First, it’s not guaranteed that someone will win. It’s very possible that there won’t be any winning tickets. The more tickets get sold, the greater the chance that someone will win, but that also increases the chance that the jackpot will be split. And remember that the advertised number isn’t the actual jackpot. Unfortunately, we’ll have to make educated guesses for some of the numbers. Suppose they advertise that the jackpot this week will be $18 million and we think there will be 30 million tickets sold. Suppose this is a standard $1 “pick six numbers from 1 to 49” lottery, with 13,983,816 unique combinations of numbers. Let’s call that last number n; you’ll see why in a minute. All things being equal, any ticket has 1/n chance of winning. But, assuming that someone wins, the best guess for how many winning tickets there will be is 30 million divided by n. The jackpot will be divided by this number, which means we’ll actually multiply the jackpot times n over 30 million. And lastly, remember that the actual jackpot is only about 60% of the advertised jackpot. And we need to multiply all this by the probability that someone will win. Given 30 million tickets, I’ll estimate that to be 75%.

EV = (75%)x(1/n)x(60%)x($18 million)x(n/30 million)-$1.00

Notice there’s an n in the numerator and denominator, so the n’s cancel. Same goes for the “million”. That just leaves…

EV = (75%)x(60%)x($18)x(1/30)-$1 = $.27-$1.00 = -$.73

This is a really bad expected value. It’s negative (of course) meaning the odds are tilted against the players. On average, each ticket costs $1.00 and loses $.73. That’s a huge profit for the house.

Well, let’s suppose that no one wins the jackpot this week and it rolls over to another week. Now they’ll another 30 million tickets and the advertised jackpot is $36 million.

EV = (75%)x(60%)x($36m)x(1/30m)-$1 = $.54-$1.00 = -$.46

This is better, but it’s still a large profit for the house, and that’s on top of all the profit they made last week when there were no winners at all.

Let’s take it one more step. Suppose that once again there are no winners and it rolls over again. This week they advertise the jackpot to be $72 million. Why such a big jump? Because they expect to sell more tickets! This week there will be 60 million tickets instead of just 30 million. Now it’s 90% certain that someone will win.

EV = (90%)x(60%)x($72m)x(1/60m)-$1 = $.65-$1.00 = -$.35

The house still expects to make a 35% profit this week, on top of all the profit they made last week and the week before. The only way that your EV becomes positive is if there’s a rollover followed by a week where very few people buy any tickets. But they’ve convinced everyone to buy the tickets because $72,000,000 sounds great.

Now, let’s use the real-world numbers from last week’s Powerball Lottery. The jackpot was advertised as $1.5 billion and they sold 371 million tickets.

EV = (85%)x(60%)x($1586m)x(1/371m)-$2 = $2.18 – $2.00 = $.18

Amazingly, we’ve actually found a game with a positive expected value, which means it favors the players (slightly). Players could expect a 9% return on their investment. This leads to another question. If you could buy one of every single ticket, would it be worth it? First, consider that n=292,201,338 for Powerball, so you’d need to buy that many tickets. And you’d have to hire an army of 120,000 people to help you buy them, which would cost around $50 million to pay their salaries for one week, bringing your total investment to $634 million. You’d be increasing the number of tickets sold to 663 million, and the jackpot would go up another 200 million or so. Also, you’d be guaranteeing that there would be at least one winner.

EV = (100%)x(100%)x(60%)x($1786m)x(292m/663m)-$634m = $471m-$634m = -$163 million

So that would be a really bad business plan. It’s not smart to invest $634 million when you expect to lose $163 million of it. The EV changed when you altered the game by buying so many tickets.

Anyway, what actually happened last week is there were 3 winners, so each of them got $328 million (Present Value), which isn’t nearly as big as $1.586 billion but it’s still huge. But is it really all that great? Will it make you happy? Will it solve your problems? That’s the topic for part 3.

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