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.

Barbara, in reality your 1,2,3,4,5 and 6 combination of numbers do not have the same chances as other random numbers. Your considering “odds” but not taking into consideration “probability”. Yes, the combination of numbers has the same “odds” as any other set of numbers – 1 in 292 million. However, the probability of hitting consecutive numbers is extremely much less probable than hitting random numbers and thus offer a highly decreased probability.

To simplify this, let’s consider a coin toss: you flip a coin and know that the odds of flipping “heads” is 1 in 2 (.5), in fact every time you flip the coin you have the same odds regardless of how many times you flip the coin – it’s always 1 in 2 that it will come up heads. But the laws of probability dictate that given enough flips over time the heads vs tails will pretty much even out a bit. In other words, you can keep flipping heads say 3, 4, or 5 times in a row or more but eventually it is going to land on tails. So every time you flips “heads” consecutively, the probability that the next flip will be tails increases with each consecutive flip of heads, even though the odds remain the same. If it doesn’t then an unfair advantage exists.

So the odds of hitting the Powerball with 1,2,3,4,5 and 6 have the odds of 1-292 million, same as any other number combination; but the probability of hitting consecutive numbers is astronomically improbable and gives you a much lower probability than someone who randomly chose their numbers.

Carl, you’ve committed the gambler’s fallacy:

“The gambler’s fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature). In situations where what is being observed is truly random (i.e., independent trials of a random process), this belief, though appealing to the human mind, is false.” — Wikipedia.

Applying a rational view to the lottery as a whole, you can see that overall the rate of return is less than 1. In analogy to gambling the lottery has a house edge – the proportion of the ticket needed to run the game, and the proportion paid out to “good causes”.

Expected rate of return applies in other endeavours too. I was taught it in the context of project planning in business to make estimates of whether a project was likely to benefit the organization. Something that ought to be taught as much as how to cost a project or layout a Gannt chart.

I would like to argue against the idea that gambling, lottery, and other endeavors with EV<0 are inherently "unfair" because of the EV<0 aspect. People play roulette because they know that it's -5% loser, not because they're expecting to lose, but because the marginal utility of all the other things that go along with winnning (emotion, prestige, the feeling of winning) vastly outweigh the small price they pay in house fees or rake. It's an individual calculation, sure, so it's much harder to quantify than the absolute expected value, but safe to say individuals don't view roulette as unfair. They simply have larger criteria for "fairness" than how much money they win.

In order for roulette and the lottery to be "unfair", I would argue that it happens not when the games have EV<0, but when the rules are unknown or change after the participants initiate. For exampe: suppose your roulette table didn't pay out 2:1 on the colors or 36:1 on hitting an exact number. Instead, this table pays out at the table manager's discretion, based on how many people are playing, whether the casino has been losing money recently, or any other arbitrary aspect. In this situation, you would still have the EV<0 (because the house wins on 0 and 00), but the fairness of the game has been compromised because the rules are unknown. You might play Red all night; the first time you win, you get 2.25 because the table manager likes you. The next time, 2.50, because he wants to keep the lady next to you happy, too. The next time, you get 1.75 and the lady next to you gets 3.50. This, then, we would describe as unfair, and would storm off to the casino owner for complaints about shenanigans.

The lottery may be either a "fair" game with EV<0 because it has strict adherence to the rules (as in roulette) or it may be an "unfair" game, with EV0 would somehow make the game fair again. I disagree. I don’t think the lottery is unfair because people expect to lose money. I think the lottery is unfair because I don’t have any way to verify whether the officials have followed the rules (if there are any) about how large the jackpot should be based on the # of tickets sold. We all see the numbers continue to rise as more and more hopefuls purchase their tickets to financial freedom. But we are left at the mercy and discretion of the lottery officials (similar to the roulette table manager above) who could give us some, most, or all of the amounts they are supposed to. We simply have to trust them, and that’s the definition of unfair, in my mind.

None can doubt the vetaciry of this article.

In Michigan, scratch off lottery tickets seem to fly off the shelves. It must have something to do with instant gratification.

The lottery is “change your life” gambling. The outlay is small, the potential prize huge – that tends to negate the fact that the odds against you winning are also huge. That’s why it attracts poor people. That’s why I buy the occasional ticket :).

Many thanks for your article, Barbara.

“Random processes tend to run in streaks” — correct. Poisson clumping (aka Poisson bursts) and queuing theory explain such things as, after waiting ages for a bus or a taxi, three arrive together. A long sequence of random data that does not contain streaks of easily recognisable patterns is extremely unlikely to occur. Humans are hardwired to find patterns, which is why we frequently mistake correlation for causation, often leading to the formation of superstitious beliefs.

The other thing that surprises many people is when a lottery draw repeats the same set of numbers as a previous draw. The reason for this unexpected happening is the same reason as the birthday problem aka the birthday paradox.

As others have said, the best set of numbers to play are those chosen by the least number of other players. Always remember that a priori probabilities become increasingly meaningless/inapplicable as the sample size reduces below 30. For each player (sample size = 1) the outcome is either win or lose: for the winner(s), the investment was worth making; for all the losers, it wasn’t (other than for the enjoyment of taking part).

An LA Times Business columnist says people “pay less attention to the probability of a given event than the consequences if it takes place,” whether it’s a lottery win or a terrorist attack.

But I dunno, the risk of horrific death doesn’t stop people from smoking or from texting while driving.

If you don’t win, you can fantasize that the revenues will fund schools as advertised, instead of being shuffled around and wasted as usual.

To a normal person, there’s no qualitative difference between $900 million and $300 million. It’s all a boatload of money, more money than you know what to do with. Expected value is an abstraction. You’re not going to play the lottery a trillion times for the law of large numbers to take effect and yield the small expected value.

In reality, people maximize the expected utility, not expected winnings. For example, would you rather have a 100% chance of winning a million dollars or a 51% chance of winning two million dollars? I’d go for the sure thing. You can model utility as the logarithm of money, so utility increases by one unit when the money is doubled.

Seven-time lottery winner Richard Lustig has been doing the talkshow circuit again, giving advice on how to win the lottery, like don’t use randomized numbers, and buy as many tickets as you can afford. He’s been interviewed on CNN, ABC, etc.

Apparently, he never says how much money he’s spent or if he’s even a net winner.

He inadvertently gives some good advice, like don’t pick birthdays, but not to reduce the risk of splitting the prize. He actually thinks that this affects the chance of having the winning numbers.

CNN was shamed into removing the video of his interview in 2012, but he’s back at it again, featured on Yahoo news and freakin Forbes.

https://en.wikipedia.org/wiki/Richard_Lustig

I’m confused about how you concluded it’s rational based on expected value. As the charts on the below link demonstrate, the expected value for this Powerball lottery was between -$0.25 and -$1.49 (depending on # of tickets sold, which determines the the odds of a split jackpot).

Those are the figures assuming the winner would take the lump sum (which almost all do) and lives in a high-tax state. For those living in states like FL and CA without lottery taxes, the expected value is a little higher, but still negative. As the charts show, even with the annuity option, expected value is negative.

How is a bet with a negative expected value rational? Or can you point out an error in those calculations?

Thanks!

http://www.businessinsider.com/powerball-lottery-expected-value-jan-13-draw-2016-1

I hope you didn’t really mean to write “Random processes tend to run in streaks” because that is wrong. To run in a streak, a process would have to “remember” its prior history, and as such would not be truly random. Humans may THINK they observe streaks in a random process, but that is a issue of human perception and not of statistics.

Agreed, my language is not accurate. I apologize.

There is a kind of psychological way that may make the lottery ‘fair’.

If your rather poor, but still have a spare dollar now and then, the payout of the lotter may seem almost infinite, i.e., life changing against the dollar that even for you is not really significant.

An occasional bet may cheer you up.

However, it you get hooked by the ‘intermittent reinforcement’ of the smaller payouts, it can in reality make things much worse for you.

-Traruh

The real “value” of a lottery ticket is the dreams: If I buy a ticket I can dream about what I’d do with the money. “You can’t win if you don’t play,” is a common trope. But with only one chance in three hundred million of winning, my chances of finding the winning ticket on the sidewalk are only slightly less than my chance of winning on a ticket I buy myself.

Therefore, instead of buying a ticket, I dream about what I’d do with the money if I found the winning ticket on the sidewalk.

I’ll admit, however, that greed is a powerful motivator, and I did look at the lottery machine as I walked past it at the grocery store this morning. I’ll admit further that the main reason I didn’t throw away $2 on it was that I’d be embarrassed to be seen as “one of those idiots.”

Daniel- I agree with your comment about the real value is in the dream.

Here’s an idea to get the most for your dollar when buying dreams. Some lotteries have an advance play feature so that you can choose your numbers for a draw for ten (or more) weeks in advance. Also, some lotteries allow a winner to claim up to a year after the draw. So buy one ticket for the draw ten weeks in advance, then don’t check your ticket until a year after the draw. That way, your dream only costs a dollar (or two for Powerball), yet lasts 62 weeks!

Yeah, that’s the tietkc, sir or ma’am

Buying insurance is not a fair bet, because the insurance companies take their cut, and the total return is less than 1. But most of us buy insurance anyway, because we do not want to take a chance of catastrophic loss. Lotteries are generally not a fair bet, but many of us buy tickets anyway because we are willing to blow a dollar for the chance of incredible riches.

Times are chginang for the better if I can get this online!

Barbara- Thank you for writing this article. It really clarifies the concept of expected return.

I remember a series of advertisements for the California lottery many years ago. Each ad featured a celebrity and their favorite numbers. One had Steve Wozniak indicating his numbers were 1-2-3-4-5-6 and saying something to the effect that, Why not? It’s as likely to win as any other combination. I am unable to find that ad via internet search, but do see others mentioning it. But also, I see this article from a couple years ago from the UK’s Daily Mail saying that 1-2-3-4-5-6 are the numbers to avoid as about 10,000 people choose those each week. http://www.dailymail.co.uk/news/article-2301360/The-Lotto-numbers-avoid-Going-1-2-3-4-5-6-bring-tiny-windfall.html So we have psychology running both with, and counter to, probability. Interesting! I think the commenters suggesting picking numbers above 31 to avoid calendar dates and thus more likely to avoid splitting the pot have something there.

Also, an interesting calculation would be to determine the chances of two or more having to divide the jackpot. It seems this information should be ‘gettable.’ If the percentage of the cost of each ticket that funds just the jackpot is available, could you simply subtract the last jackpot ($900,000,000) from the current jackpot ($1,500,000,000) and divide by that percentage to yield the number of tickets purchased for this draw? I may not have it exactly, but I think I’m on to something. Once armed with that number, there must be a formula for determining the chances of any particular combination being selected more than once, but it’s beyond my maths.

“I think the commenters suggesting picking numbers above 31 to avoid calendar dates and thus more likely to avoid splitting the pot have something there.”

As do I.

Thank you for doing that research!

Playing the lottery is only “irrational” if you take it as a given that the only rational financial goal for a person to have is to maximize their financial well-being. With that premise, having $1 is better than having a <1/100 million chance at $100 million. But what if a person's goal is to be incredibly wealthy, to own mansions and private jets and fancy cars? For most people, the lottery is the only path to that goal — no amount of saving or prudent investment will get them there, they don't have skills that can be leveraged into great fortune, and any other type of gambling would require them to risk destitution. It's worth paying a premium for the only shot at your goal. You can say being incredibly wealthy is a stupid and irresponsible goal, but that's a question of values, not of math.

Well said and this possibly explains the impulse behind some purchases. But it doesn’t chance the odds.

This article is obviously mathematically sound, but ignores opportunity cost. The opportunity cost of buying a lottery ticket in most cases is infinitesimally small.

If I buy a ticket for say $2 every time the jackpot hits $500 million, it would be correct to say that this is still a sucker bet, but most people don’t miss the $2. It’s equivalent to taking the change out from under your couch cushions and throwing it in the garbage. Except there is an exceedingly small chance you could win $500 million by doing so. (Or in this case, $1.3 billion.)

That’s not to defend lotteries in general, just pointing out that a lot of people (like me) know the odds are terrible but it’s not a big deal throwing away two bucks and playing the game.

Wrong, John Haigh. What makes the ‘winning the lottery’ fantasy enjoyable is the presence of a possibility – however remote – that it could happen. The ‘rich relative’ and ‘sudden million-dollar inspiration’ fantasies clearly have no chance. I confess I buy a ticket maybe a half-dozen times a year; probably less.

ACW: Then just fantasize that someone will play for you and is honest enough to pay you if it hits. It’s fantasy!

Henry – I can create plenty of ‘get rich quick with no work’ fantasies without wasting money on a lottery ticket. Rich relatives I’ve never heard of, a sudden inspirational invention, etc.

Lottery – a tax on people who are bad at maths.

Agreed, I always imagine becoming some tech genius business man. While work would be needed, you’d at least be a contributing person, and have millions of dollars!

Math. You don’t shorten Economics to Econs, do you?

While the math may say that there is an equal chance of winning with 1,2,3,4,5 plus 6 for the powerball, a quick review of the historical record suggests that your odds would be better with 65, 66, 67, 68, 69 and whatever you pick for the powerball. Going back to 1997 the lowest high white number ever drawn appears to be 19. It is rare to see a high white number below 30. Why this is the case is hard to fathom since the numbers are not computer generated, but are physically randomized in a machine, but that is the way the game has played out since its inception.

I have to agree about 1,2,3,4,5,6 – or indeed, any plausible pattern. The vast majority of people _won’t_ pick on that basis – but it only takes _one_ other person to pick those numbers to halve your expected return. Since all numbers are equally likely to win, the only reason for choosing a number is to make it less likely that you’ll share it.

Rollovers do indeed increase your expected return, because all the people who bought tickets the first time around are out of the draw and you’re betting against their money.

One could do some fairly elaborate analysis about what strategy is likely to give the best return, but it’s probably wasted effort, given the extreme unlikeliness of winning.

What a cop out. You gave no analysis of the probability of sharing the pot and the expected outcome from that or accounting for the smaller prizes or which numbers are least picked.

You also should also make a consideration to the expected happiness outcome, as winners of the lottery are often not happier afterwards.

I’d like to know what the optimal amount to win is for optimal happiness.

I’m guessing somewhere south of 10 million.

Otheus, where id you learn statistics? The total number of possible draws is far higher: 69 pick 5 times (something, 54 or 69) This is a number in the gazillions.

There were only 26 Powerball numbers, so it’s (69 choose 5) times 26, which is 292 million. That’s why the chance of a Jackpot is 1 in 292 million. At $2 per combination, all possible combinations would cost $584 million.

Yes it is 1 in 292 million, but you have excluded all the other ways of obtaining a return on the investment: “Overall odds of winning a prize are 1 in 24.87”

https://en.wikipedia.org/wiki/Powerball#Prizes_and_odds