Archive for February 2010

## Wise Men Without Hats

I am so used to wise-men puzzles about hats, that I was pleasantly surprised when Leonid Makar-Limanov gave me a wise-men puzzle that didn’t include them.

A sultan decides to check how wise his two wise men are. The sultan chooses a cell on a chessboard and shows it to the first wise man. In addition, each cell on the chessboard either contains a rock or is empty. The first wise man has to decide whether to remove one rock or to add one rock to an empty cell. Next, the second wise man must look at the board and guess which cell was chosen by the sultan. The two wise men are permitted to agree on the strategy beforehand. What strategy can they find to ensure that the second wise man will always guess the chosen cell?

If you are stuck, there are many approaches to try. You can attempt to solve the puzzle for a board of size 1 by 2, or for a board of size 1 by 3. Some might find it easier to solve a version in which you are allowed to have a multiple number of rocks on a cell, and the first wise man is permitted to add a rock to a cell that already contains rocks.

## Women in Numbers

This year I am again on the organizing committee of the Women and Mathematics program at Princeton’s Institute for Advanced Study. Our subject is “p-adic Langlands Program.” It is a fashionable, advanced and very influential program connecting number theory and representation theory.

We invite undergraduate students, graduate students and post-docs to apply. In 2009-2010 the Institute has been running a special year in Analytic Number Theory. That has brought many number theorists to the institute already, so there will be a lot of people to talk to.

Last year I promised to hold a math party during the program. But I had to cancel it due to a scheduling conflict with George Hart’s ZomeTool Workshop. I am planning a party this year. Either way, we’ll have fun.

If you want to learn about the Langlands program, to spent time on the beautiful grounds of the Institute, to eat in one of the best cafeterias around, and to make new friends with other women interested in number theory, then please apply. The application deadline is February 20.

## Office Lottery Pool

Suppose you want to increase your chances of winning the lottery jackpot by pooling money with a group of coworkers. There are several issues you should keep in mind.

When you pool the money and you hit the jackpot, the money has to be split. If you bought 10,000 tickets and the jackpot that you win is \$100 million, then each ticket is entitled to a mere \$10,000. Your chances of hitting the jackpot in the first place are 1 in 17,500 and you’re not going to get rich off what you win.

Perhaps you’d be satisfied with a small profit. However, as I calculated in my previous piece on the subject, even if you include the jackpot in the calculation of the expected return, the Mega Millions game never had, and probably never will have a positive return.

Despite this fact, people continue to pool money in the hopes of winning big. However, there are more problems in doing this than just its non-profitability.

Consider a scenario. Your coworkers collected \$1,000 to buy 1,000 lottery tickets. You give the money to Jerry who buys the tickets. Jerry can go to a store and buy 1,005 tickets. After the lottery he checks the tickets, takes the best five for himself and comes back to work with 1,000 disappointing tickets.

It is more likely that Jerry is cheating or that he will lose the tickets than it is that your group will win the jackpot. But there is a probabilistic way to check Jerry’s integrity. According to the odds, every 40th ticket in Mega Millions wins something. Out of 1,000 tickets that Jerry bought, you should have about 25 that win something. If Jerry systematically brings back tickets that win less often than expected, you should replace Jerry with someone else.

There are methods to protect your group against cheating. For example, you can ask another person to join Jerry in purchasing the tickets, which they then seal in an envelope that they both sign.

Alternatively, you yourself could be the person responsible for buying 1,000 tickets. How would you protect yourself from suspicion of cheating? The same way as I mentioned above: bring along some witnesses and have everyone sign the sealed envelope.

The most reliable way to prevent Jerry from cheating is to have him write down all the ticket numbers and send this information to everyone before the drawing. This way he can’t replace one ticket with another. But this is a lot of work for tickets that are usually worth less than the money you collected to buy them.

But there are other kinds of dangers if you use this supposedly reliable method. If you bought a lot of tickets the probability of winning a big payoff increases. Suppose Jerry publicly locks the envelope in a desk drawer in his office. If one ticket wins \$10,000, and everyone knows all the ticket combinations, suddenly Jerry’s desk drawer becomes a very unsafe place to keep the tickets.

Scams are not your only worry. You shouldn’t buy the same combination twice — whether picking randomly or not. You really do not want to waste a ticket and end up sharing the jackpot with yourself.

You cannot change the odds of hitting the jackpot, but you can change the odds of sharing it with others. Indeed, there are people who do not buy random combinations, but rather pick their favorite numbers, like birthdays. You can reduce the probability of sharing the jackpot if you choose the combinations for your tickets wisely, by picking numbers that other people are unlikely to pick.

Still want to try the lottery? If you feel a need to throw your money away, instead of buying lottery tickets, feel free to donate to this blog.

## The Expected Return on Lotteries

In one of my previous pieces, I discussed returns on the Mega Millions lottery game, assuming that you buy a small number of tickets. In such a case winning the jackpot has zero probability. So I argued that if you want to estimate the profitability of the lottery as an investment, you have to remove the jackpot money from the calculation.

Today I will discuss what the formal expected return is. That is, I will include the jackpot money in the calculation. Since I argued against including the jackpot in my last article, you might wonder why I’ve then turned around to look into this.

I think this mathematical exercise will be fun. Besides, on a practical note, it is useful to know when the formal expected return is more than 100%, because then it might make sense to pool money with other people. Keep in mind though that if you want a chance to hit the jackpot, the total number of tickets you buy must be really big. For example, even if you manage to pool \$10,000 for tickets, your probability of winning the jackpot in Mega Millions is only one in 17,500 — still minuscule.

If you buy only one ticket, you’ll lose. If you manage to pool a lot of money and the probability of the jackpot becomes noticeable, that is, non-zero, could the jackpot be large enough that the lottery becomes a good investment?

For this calculation, I’m still assuming that you buy a relatively small number of tickets. If you buy millions of tickets the calculation is slightly different, and I will write about that later.

You might think that when the jackpot is bigger than the odds, it makes sense to play. I am discussing the Mega Millions game, where the odds of winning the jackpot are one in 175 million. So if the jackpot is more than \$175 million, then it is profitable to play. Right?

Wrong. As I mentioned in my previous piece, after reducing for taxes, you get about 16% of your money back through smaller payouts. Hence, you need to recover the other 84% through the jackpot. So the jackpot should be more than 175*.84 = 147 million dollars. This sounds even better. Right?

Wrong. No one receives the jackpot. Winners can chose to immediately receive the lump sum, which equals the money lottery organizers have actually set aside for it. Alternatively, the lottery organizers can invest the lump sum and give winners a yearly distribution over many years, the total of which will equal the jackpot.

Suppose for simplicity the lump sum is half of the jackpot. That means we need the jackpot to be \$294 million (\$147 x 2). Right?

Oops. As usual, we forgot about taxes. To exacerbate your pain, I have to add that the winnings are taxable. Suppose you have to pay 30% from the jackpot. That means the jackpot needs to be \$424 (\$294/0.7) million in order to justify pooling money. OK?

We haven’t seen jackpots that big yet. But neither have we finished the calculation. There is a probability that you might have to share the jackpot with other winners. To calculate this probability, we need to calculate the number of tickets sold. That means, your expected return depends not only on the size of the jackpot, but also on the number of these tickets.

But even if you know the number of tickets sold, we cannot calculate the expected returns precisely because people don’t always buy tickets with random combinations, but often pick their own numbers.

When the jackpot is large people start buying tons of tickets, so we can expect that many of them buy quick-picks. Let us assume for now that the vast majority of people do not choose their own numbers, but buy tickets at random. Suppose 200 million tickets were sold. That is a very big number. Last time that many tickets were sold was when the jackpot was \$390 million in March 2007. By the way, that was the largest jackpot ever.

In order to finish the calculation, we need to establish the probability of several winners, given that 200 million random tickets were sold:

Number of winners Probability
No winner 0.3204
One winner 0.3647
Two winners 0.2075
Three winners 0.0787
Four winners 0.0224
Five winners 0.0051
Six winners 0.0010

From here we can calculate the adjustment coefficient, that is, the proportion of money you are expected to get from the jackpot given that there are 200 million players in the game. The coefficient is calculated from the table above as (0.3647 + 1/2*0.2075 + 1/3*0.0787 +1/4*0.0224 + 1/5*0.0051 + 1/6*0.0010)/(1 - 0.3204), and is equal to 0.7379. We need to divide our previous figure of \$424 million by the adjustment coefficient. The result is \$575 million.

Given that a \$390 million jackpot attracted more than \$200 million in tickets, we can expect that the \$575 million jackpot will make people completely crazy and attract even more money. So I do not anticipate that the Mega Millions game will ever have a positive formal expected gain. My conclusion is that not only is there no financial sense in buying a single lottery ticket, but also none in pooling money.

Of course, you can buy tickets for non-financial reasons, like pumping up your adrenaline. In any case, I showed you the method to calculate your expected return, or, more appropriately, your expected loss.