**Still Here**

The unthinkable has happened: neither I, nor most certainly you, have won the lottery.

Notice I did not use the word “inconceivable”, but merely “unthinkable.” For while it was surely conceivable (and more) that I (we) would not win, it was not a thought that repaid thinking. One of the main purposes of the lottery after all is to provide cheap, harmless entertainment, to allow one to indulge in legitimate fantasy (as I did yesterday).

For the record, the winning numbers were: 2, 4, not mine, not mine, not mine, and Mega Ball not mine.

Now, I noticed some odd things in the mania which led up the drawing. The most prominent is the number of people who were careful to tell us that they did not buy a ticket, news which they thought imbued them a certain level of sophistication, of announcing *they* at least were above the fray.

**You Are More Likely To Die Horribly**

The lottery, particularly the jackpot-style gambles like Mega Millions, are oft called “idiot taxes” and with some justice. The chance of winning (for buying one ticket) is 1 in tens and tens and tens and etc. of millions, and the payout is usually low. It really is like throwing money away for ordinary payouts—but not necessarily for huge jackpots. The irony is that it takes a large number of ill-advised bets—weeks and weeks of them—to build a jackpot to the level for which it makes sense to play.

For *this* jackpot, it *did* make sense to play, at least for most folks. More on this in a moment.

More than a few sites reported that you were more likely to get run over by a bus than to win the lottery. Or there was a higher probability of falling on your head and splitting it open (I saw this somewhere but can’t rediscover where), or to die of a horrible disease, or to be struck by lightning, and on and on. One gruesome calamity after another.

Why must the comparison always be with something evil? Why not say it is probable you’ll get your book published, or that you are seven times more likely to be discovered by Hollywood, or that there is much more of a chance you will be the guy to rescue a box of puppies from a raging flood?

Well all of these are the wrong numbers, utterly irrelevant calculations. Your buying or not buying a lottery ticket does not, in an any way, affect the chance you will be struck by lightening—unless you were stepping into a bodega to buy a ticket and fortuitously avoided a storm. It’s the same with any other good or bad thing which might happen to you: unless that thing has to do with the money you spend on tickets, the numbers are meaningless.

**Fair Games & Expected Values**

I spent one dollar; I bought one ticket. I figure this was a fair trade for the chance at the money I might have won and for the fun I had in writing up the fantasy of what I would have done with the winnings. I’ll note that the “calculation” which justified this outlay was valid for me, but perhaps not for you. For a guy who is struggling to find money for food and rent, making the gamble is ill-advised.

As to the size of the actual payout, we can use the rough rule of thumb to divide by 3 the advertised jackpot. This is because the government will step in and take their enormous bite, and you do not actually win the jackpot but the cash which would buy a twenty-five-year annuity (for the Mega Millions). The cash for this lottery will be about $280 million.

Now there is in statistics a calculation called “expected value”, which does not mean what its English words would imply. It is simple to produce, however. It is the probability of the event times the value of the event, summed over all possible events. For the lottery, there are only two events: winning the (modified) payout and winning *nada*. So the “expected” value is p * $280 million + (1-p) * $0 million, where “p” is the probability of winning, which for the Mega Millions is about 1 in 176 million. Thus, the “expected” value is about $1.60.

Statisticians call a gamble “fair” or even desirable if its expected value is larger than or equal to the cost of the gamble. $1.60 is greater than a dollar, so the gamble is “fair.” But that figure also means that you should have been able to buy lottery tickets from a vendor and then gone out onto the street and sold the tickets for $1.59 each, since anybody who paid this price still “expects” to win a positive amount for each ticket. As the kids say, good luck with that.

One of the problems with the “expected” value calculation is that it treats money in a way differently than human beings. Doubling your fortunes from $1 to $2 is much different to a human being than is going from $50,000 to $100,000, while both of these are the same to the soulless mathematics. To get around this, economists turn money into “utility”, an attempt to turn the calculations into something meaningful. The difficulty is that everybody has a different “utility”, so this is merely giving the problem a label and hoping it goes away.

The solution is that everybody must decide for himself whether the gamble is worth it or not. For me, the payout was not just cash, but valuable fantasy points which I spent freely. And I repeat: for me, the gamble was fair.

**More Than One Winner**

Word is that the money will be split three ways, for there were three winning tickets. That there would have been multiple winners was easily foreseen, as lottery officials have compiled data which show that the number of tickets sold is roughly proportional to the size of the jackpot, and this was the largest jackpot in history. Just how many winners there would be nobody knows in advance, but we could use the data to estimate the chance that there would be just one winner, just two, etc.

Any gambler worth his spurs would have figured this into his “expected” value calculations. If he figured there would have been three winners, then he would have divided the $280 by 3 and produced an “expected” value of about $0.53, which according to traditional wisdom makes the gamble “unfair.” But that’s only if you treat money linearly, which again human beings do not. I knew of the possibility of multiple winners, yet I still figured the gamble fair (for the reasons given above).

**Strategy**

There is folk wisdom which says that you should not pick “common” numbers, because common numbers are by definition also picked by other gamblers. For example, there are many tickets sold for the sequence “1 – 2 – 3 – 4 – 5 – M6”, a set which has the benefit of being easily remembered. Too, that sequence has just as much chance as any other sequence of arising, so just because you pick novelty numbers does not make it less likely it will arise.

*If* the bouncing balls do spit out the sequence “1 – 2 – 3 – 4 – 5 – M6” you will share the jackpot with the other comedians who bought this set of numbers. And since that is so, your “expected” winnings are consequently lowered, just as they were for yesterday’s drawing.

But this calculation, while valid, is the wrong one. Just think: if you pick an uncommon sequence and eschew the sequence “1 – 2 – 3 – 4 – 5 – M6” but this common sequence comes up, then how much do you win? Nothing. It’s true that *before the drawing* your “expected” payout was higher for your uncommon set than the “expected” value for the saps who picked 1–6. But you can’t spend “expected” money (if you think so, before the drawing try selling your ticket for more than a dollar). And you still lost. You split nothing with yourself, while the comedians split something. And something is more than nothing.

The real strategy, since nobody knows what sequence will arise and all sequences are equally likely, is to either pick numbers which you can easily remember, or let the machine do it for you since that is faster. But do wait until the jackpot pushes past a couple of hundred million. Anything less makes it less likely you’re gambling for entertainment and instead engaging in some perverse investment strategy.

Had you placed a bet would the ripple effect of your movements have spread so as change the draw?

Many years ago when the Mafia ran the only “lottery” they had a numbers game that people could play. My parents played the same numbers and would win about every 4 or 5 months. In fact the numbers they played payed a lower payoff. This is because of an anomoly you might be interested in. The Mafia needed a number that would be random and available to everyone. They used the last 4 digits of the total receipts from the local race track. The number would be printed in the paper. Here is the anomoly; my parents always uses “1010” and because the number was selected from cash reciepts “1” and “0”s came up more often. Any statistical story you can weave into this???

The reason you can’t sell your ticket before the drawing for more than $1 is because there are plenty of lottery retailers who will sell an equivalent ticket for exactly $1. Imagine that the retailers all stopped selling several days before the drawing. Do you think you could get more than $1 for your ticket then?

Eric,

This is a good point, but it weakens when you imagine that you could drive to some remote spot where there are no lottery retailers. You might get more than $1 for your tickets, but I’d imagine you wouldn’t see $1.59.

I’m clearly missing something in your strategy discussion. Suppose person A picks 1,2,3,4,5,6 while person B picks less common numbers. One of three things will happen: A will win, B will win, or neither will win. The probability of the first two events, while incredibly small, are equal, but if B wins he will get more money than A. How can this not mean that B made a better bet than A?

Correct me if I am wrong, I think the jackpot is determined in a way so that those states will not lose money. So, the lottery is not a fair game. I guess itâ€™s fair because every ticket has the same chance of winning the jackpot.

Anyway, people who buy lotto tickets probably donâ€™t think about whether itâ€™s a fair game and what the expected payout is. Itâ€™s hard to resist the lure of the enormous jackpot and possibly the lure of the winning for matching five or fewer numbers (your calculation of the expected payout doesnâ€™t take these events into consideration). Even the frugal Mr. JH bought 10 tickets yesterday.

JH

There is a big difference between buying no ticket vs. one ticket. (Fantasy points)

There is no significant difference between buying 10 tickets vs. 1. (That’s gambling)

BTW. I noted that the Mega number was 23 while one of the 5 numbers was also 23.

I believe most number pickers do not duplicate their numbers. If there had been no quick-picks, there probably would not have been a winner. Although, the buyers of large numbers of tickets, like JH and that guy who bought $2600 worth (also a gambler), probably did not personally pick their numbers.

Cheers

Anne

This is indeed as stupid as it seemed on twitter yesterday.

I can’t put it better than Eric has above.

It is possible that a 1-in-ten-million chance of winning five million dollars might be worth more than a dollar to you, since you’ll never miss the dollar, but five million would change your life. That’s an example of the utility argument, and I agree that the fact of the “expected value” being less than the ticket price doesn’t necessarily make it a bad thing to buy the ticket.

But isn’t a 1-in-ten-million chance of winning five million dollars better than a 1-in-ten-million chance of winning 1 million dollars? You don’t have to “treat money linearly” to prove that, you just have to treat it monotonically.

Sure, after the numbers are drawn, the best numbers to have picked are whichever numbers came up, whether that’s 1-2-3-4-5-6 or 8-18-23-30-33-41. But you make your choice before they’re drawn, and you can’t influence your chance of winning by your choice of numbers, but you can influence the amount that it is possible for you to win.

According to what I read some time ago, literally thousands of idiots pick 1-2-3-4-5-6 in major lotteries. So it’s impossible to win a million with those numbers, it’s bound to be split too many ways.

Oh, and the reason you can’t sell on your ticket is because any potential buyer would assume you’re up to something: either the ticket is forged, or invalid in some other way. The cost of establishing that the ticket is really what it purports to be is more than the price.

JH,

“fair” depends upon what you mean. The utility thinkers have it right. I know some people who spend $1000/mo. in a bar with an expected return of $0. Spending $10 on an infinitely small chance of some monetary return is still many times larger than the guaranteed return of $0 for not playing. And when compared to the $1000/mo. for entertainment it’s like tossing a penny into a wishing well.

No one would spend any money on anything if they truly get nothing out of the transaction. How much are hopes and dreams worth? Even with extortion the payer gets something out of the transaction if only a (perhaps false) belief in continued well-being.

Anne,

There’s little difference between spending $1 on a chance to win a huge jackpot vs. $10 on 10 chances. The amount spent reflects what the chance of winning is worth to the individual. Nearly everything a person does is some form of gambling. The act of getting up and going to work increases one’s chances of being killed in a transportation accident yet many find the gamble worth the price. Assuming your price is universally applicable is unwarranted.

Gone with the wind,

This may be related to Benford’s (Newcomb’s) Law, although it may not apply for the situation as you describe it.

http://en.wikipedia.org/wiki/Benford's_law

DAV,

Whether a lottery game is fair (fairness) is different than how one decides (utility, satisfaction) whether to play the game. Whether one is willing to play the Mega Millions is indeed a utility-based decision. However, mathematically, a gambling game is said to be fair if your expected return is equal to what you pay in. By this definition, the Mega Millions is not fair. The fairness may or may not affect your utility function or the satisfaction you get out of a transaction.

Of course, â€œfairâ€ depends on what you mean and the scenario. Just as I said jokingly that itâ€™s fair because every ticket has the same chance of winning.

Anne, I am wondering how many tickets each of the jackpot winners bought.

JH,

agreed that the usual definition is Cost/P(succ) = Payout but that assumes the only measure of value is money.

William Sears,

Benford’s law doesn’t apply. In the lottery the numbers are nominals meaning symbols. They could just as easily be replaced with George, Jim, Harry, pictures, place names, etc. without affecting anything.

GoneWithTheWind

The phenomenon you are describing is known as Benford’s Law. Fraud examiners often use it to determine when financial data has been fabricated. Most people erroneously think that financial data should be somewhat random. In fact the leading digits follow a logarithmic distribution.

Dav,

We were referring to cash receipts, not lottery numbers.

John Vetterling,

You should read all the comments.

In the past, before the days of multi-state looteries (huh, interesting typo, I think I’ll leave it), when the pot got relatively big (but still a small fraction of today’s multi-state pots), I’d buy the same numbers multiple times. Yes, I realized that this dramatically lowered the expected value of my ‘investment’. But for me a lottery ticket was a few days of day-dreaming, and I found I just couldn’t get into the proper mood with a payout of anything less than 10x my salary for the rest of my life.

With the huge multi-state pots, 10x my salary isn’t an issue anymore, so I haven’t done the same-numbers-mulitiple-plays thing for a long time. But with all the recent stories, I think there might be yet another reason to go this route that I never thought of before . . .

I think you are correct that Benford’s seems to apply BUT it is important to restate that the lottery numbers I was referring to are on the right of the dollar amount not the left. That is it is the tens of dollars, dollar, tens of cents and cents. As I read the Benford law it applies to the higher number; the left most number.

“…you canâ€™t spend â€œexpectedâ€ money…”

Nonsense. The government does it every second of the day.

On the reselling of lottery tickets, I have had some discussions that go the other way.

If someone buys a lotter ticket for a dollar, and you offer him $1.10 as he walks out of the store. He will rarely sell it to you, even though he can turn around a just buy annother one (this is on days when the jackpot is over $500 million and there may be a line for the ticket) Now that he has invested in that ticket it is worth more than any ordinary ticket out there.

Regarding 1-2-3-4-5, I know many people who insit that that series will never come out of the hopper. I try to explain to them that, unless the system is rigged, it is just as unlikely as any other set of numbers. They just roll their eyes and walk away. However, if it did happen to be the winner, how many people would scream that there must be a fix.