With the recent Powerball mania I thought it would be a good time to talk about how to tell a sucker bet from a fair bet, and whether playing the lottery is a rational behavior.

Whenever the lottery comes up, people tend to discuss the astronomical odds of winning, holding it up as the best reason to avoid buying a ticket. However, it is not simply the odds of winning that determines if a bet is rational, at least the way psychologists define the term. A rational bet is a fair bet, and what determines if a bet is fair is something called the expected return.

An expected return is the payoff that one expects to receive—the amount, in proportion to the bet, that one expects to gain. A fair bet is one in which the expected return, in the long run, is zero. In other words, one expects to break even. If you expect to do better than break even, you are at an advantage. The odds of winning are part of expected return, but the payout-to-bet ratio is equally important in determining advantage.

In fair bet, if I bet $1, I expect to have $1 (on average) after the bet is executed; I have not gained or lost, so my return is zero. This will be the case when the odds of winning are 1 in 2 (50/50) and the payout is 2 to 1 (including the original bet). For example, if I bet $1 that a coin flip will result in a “head,” it is a fair bet if winning provides me with a payout of $2 (including my original $1). If I win I have $2 and if I lose I have $0, so on average I will end up with $1—on average I break even. This means the expected value of the bet is $1 and the expected return (gain) is zero.

The expected value is usually easily calculated in a simple game by multiplying the odds of winning by the payout. In the case of the coin flip:

Bet: $1

Payout on win: $2

Odds of winning: 1 in 2 or .5

Expected value: 2 x .5 = 1

In a fair bet, the expected value will be the same as the bet. If I bet $1 and the expected value is $1, then my expected return is zero.

Or, for a higher bet:

Bet: $2

Payout on win: $4

Odds of winning: 1 in 2 or .5

Expected value: 4 x .5 = 2

But, what if the odds of winning are much lower, as in the lottery? A fair bet can involve any odds of winning as long as the payout-to-bet ratio matches the risk, which is why even an astronomically low odds game can still be fair. For example, if I bet on the roll of a die instead of the flip of a coin, my odds of winning drop to 1 in 6. This means a higher risk, but that risk is acceptable (from a rational standpoint) if the payout ratio matches the risk at 6:1.

For example, if I bet $1 that the die will come up 3, I will need a payout of $6 to make it a fair bet:

Bet: $1

Payout: $6

Odds of winning: 1 in 6 or .167

Expected value: 6 x .167 = 1

Or, if I bet that the die will come up odd, I will need a payout of $1 to make it a fair bet, and so on.

The expected returns for casino games and other forms of gambling can be relatively simple or extremely complex, but they can be calculated for most casino games which do not involve other players. The more you look at these calculations, the more you will realize that nearly all gambling is a losing endeavor due to the house edge. That’s how casinos cover operating expenses and still make a healthy profit.

There are some very rare instances in which a casino game may provide a player advantage. For example, games such as poker and blackjack involve some skill such that it is possible for very, very highly skilled players to gain an advantage over the house (and certainly over other players, as is possible in poker). There are even games which require no skill that may give a gambler an edge: slot machines which are set to return slightly more than they take in. I will get back to those and how they relate to the lottery shortly, but first let’s talk about the house edge.

Not-So-Fair Bets

To understand understand the house edge, let’s consider the game of Roulette. Roulette is probably the easiest of the table games, but it is also probably the worst bet in any given casino.

An American roulette wheel has 38 numbered slots. Two of the slots, marked 0 and 00, are green. Eighteen of the slots are black and eighteen are red. After bets are placed, a ball is released and settles into a slot, determining the winners.

Players can bet on a number or a color. They can also place a more complicated bet such as a row of numbers, but let’s limit discussion to the simpler bets.

If one bets on black and black wins, the payout is 2:1. In other words, if you bet $1 and win, you walk away with $2 (including your original bet), like the coin flip example above. For this to be a fair bet, the odds of winning must be .5 or 50% or 1 in 2. Unfortunately, that’s not the case because less than half of the 38 slots are black (thanks to the two green slots). The odds of winning a bet on black are actually 18 in 38 or .474, not .5. So, the expected value of a $1 bet on black is actually:

2 x .474 = .947

This is more than five cents less than the bet, or an expected return of -.053 (a loss). This return is the same for almost every bet on a roulette table.

The payout when betting $1 on a number is $36. Now, it might sound great to get a payout of $36 on a bet of only $1, but keep in mind that the odds of winning are only 1 in 38, or .0263. This makes the bet unfair because the expected value is the same as that of the bet on black:

36 x .0263 = .947

The “house edge” is more than 5%, or five cents per dollar. Five cents may seem like a tiny difference, but that is merely the amount that you can expect to lose over many, many, many games. Don’t walk into a casino with $100 to play roulette for a couple of hours and expect to walk out with $94.70. It doesn’t work that way. You could walk out with $200 or more, but the chances are greater that you’ll walk out with nothing. Random processes tend to run in “streaks”, so you could be up or down by quite a lot in a short period of time. Remember, too, that you have to stop when you run out of money, but winning carries no such limitations. Therefore, if you’re winning, you’re more likely to keep playing, which increases the probability that you will swing in the other direction and give it all back.

For any form of legalized gambling, advantage to the gambler is extremely rare. In general, the house always has an edge (the expected return is less than zero) and the gambler should always expect to lose in the long run. That said, there are some cases in which the expected return is higher than zero and this is where the question of rationality comes in.

Slot Machines

Slot machines are tricky. Although there is no skill involved (except in the case of video poker machines, which complicate matters with elaborate pay tables and involve some skill), expected returns cannot be calculated because today’s machines actually allow the proprietor to set them. There are laws regulating the minimum payout proportions, but it is within a casino’s best interests to stay well above those minimums as people will stop putting money into the slots at a casino if they rarely win. In fact, some machines at some casinos might even be set to return more than what they take in. These exist (and are usually placed in high-traffic areas) to give the illusion that people tend to win when, in fact, the average across machines provides a nice profit for the casino. Of course there is no way for the individual gambler to know which machines are set for high returns, nor does such a setting guarantee that the gambler will come out ahead unless they play that same machine for several hours.

A progressive slot machine is one in which the biggest payout (the jackpot) grows over time. When someone wins the jackpot, it is reset to zero. These are usually organized as banks of several machines, each adding to the jackpot and each capable of producing a jackpot winner. In this case, your chance of winning the jackpot may be very, very small, but because that jackpot is a possibility, the expected return from your quarter (or handful of quarters, since most machines force you to play multiple coins at once for an opportunity to win a jackpot) may be significantly above zero. In other words, you may have an advantage at progressive slots. But, like the lottery, that advantage is skewed by the very high payout of a jackpot, which carries a very low probability or occurring.

A bit of a side note: one thing that people often neglect in their schemes to win big at casinos is the value of their time. Even if you found, say, a slot machine that returns 102% of your investment, chances are that you’d have to pull the handle 5000 times to make $100. If you pulled the lever every 15 seconds, that would take nearly 21 hours, or $5/hour. And there’s no guarantee you would make the $100.

Lottery

Okay, so what does all of this have to do with the lottery?

Lottery jackpots, even in the many millions, tend to be small enough that a lottery ticket is almost always a sucker’s bet. Lotteries are no different from casinos in that the game must include a house edge and the lottery’s edge is quite large.

But… there are times (like now) when the jackpot rolls over and in this way the lottery is like the progressive slot machine. The house edge is taken over time, with a jackpot payout coming from previous rounds’ bets. If a jackpot grows large enough, a bet can enter the realm of “fair”.

The odds of winning a lottery jackpot tend to be easy to calculate. For example, the odds of winning a jackpot in the current Powerball lottery are 1 in 292 million. But remember that in determining whether a bet is rational we have to consider the expected return, and that is much more difficult to calculate. When calculating the expected value of a ticket, we must include the probability of winning a lesser amount and the jackpot size. This can also be done quite easily and officials can also do a pretty good job of estimating what the pot size will be. What cannot be done easily is calculating the probability that one will have to share the pot, and this has a huge effect on the payout.

Let’s just assume for a moment that we won’t share the pot. Let’s also ignore the lessor payoffs and focus on the jackpot. Knowing the odds of winning the jackpot are 1 in 292 million and a bet (ticket price) is $2, a fair bet would involve a payout of around 584 million dollars. The current estimation for Wednesday’s Powerball jackpot is $1.4 billion.

So, using the same criteria that psychologists use to determine if a bet is rational, which is the same information gamblers use to determine if something is a good bet, buying a lottery ticket for this week’s Powerball is indeed rational behavior.

But that doesn’t mean it is wise.

The winning of a jackpot is essentially an outlier. Winning one of this size skews the distribution of outcomes in a way that makes the expected return positive and the choice look sane.

The expected return might be considerably higher than zero, but the odds of winning are still astronomically low. They do not go up with the jackpot. To put into perspective how low these odds are, if I decide to participate in the Powerball drawing this week, I will probably do so by buying a ticket with the numbers 1, 2, 3, 4, 5, and 6.

Crazy, you say? Irrational? Nope. My ticket will have the same chance of winning as any other ticket.

The internet is full of “you have a better chance” anecdotes about the lottery and for good reason. Americans spend obscene amounts of money on the lottery and the majority of that is spent by people with very little to lose. What you do need to consider is whether you can afford the $2 bet on something that is so highly unlikely to happen.

Yes, you can double your chances of winning by buying two tickets (with different numbers, of course). Then you’ll have a 1 in 146 million chance of winning. Or triple your chance by buying three tickets. But astronomical times three is still astronomical.

In the end the question of whether buying lottery tickets is rational isn’t important. The question should be whether it’s practical or wise.

Personally, I’m unlikely to join in, but I won’t make fun of you for doing so because it’s not irrational. Just… please don’t follow up your lottery ticket purchase with a gofundme plea to help you pay your bills.