Thursday, December 13, 2007

The lottery: a tax on stupidity?

It is often quipped that the lottery is a tax on stupidity (google "lottery: tax on stupidity" and you will see what I mean). I've always been bothered by this for two reasons: 1) one of the first people who said it to me was an arrogant professor who I always enjoy proving wrong, and 2) someone in my family (who has an advanced degree in mathematics) used to (and may still) play the lottery. So I figured there must be some fallacy in this statement.

I use New York's Lotto as an example, because it is a very simple "jackpot" lottery game. In this game, you choose 6 numbers between 1 and 59 (inclusive). If all 6 match, you win the jackpot. The order of the numbers does not matter. To figure out the chances of a match, you must compute the number of equally likely possible combinations (or just go the Lottos website and skip the next paragraph).

To compute these chance, we first figure the total number of combinations of 6 numbers out of 59. Suppose we choose the numbers in order. Then, we have 59 choices for the first number, 58 for the second number, and so forth until we have 54 for the 6th number. This totals 59*58*57*56*55*54=32,441,381,280 permutations in all. Since we do not care about the order, however, we need to adjust this number (consider, for example that 11,12,13,14,15,16 and 16,15,14,13,12,11 are both the same set of numbers and considered the same in the Lotto drawing). This adjustment is made by dividing by the number of possible orderings of six numbers, which is 6*5*4*3*2*1 = 720. Thus, divide 32,441,381,280 by 720, and you get 45,057,474--the number of possible combinations, of which you choose two for each $1 Lotto ticket. Your chances of winning, then, are about 1 in 22.5 million. The average jackpot is about $9 million (this jackpot amount is only obliquely referred to on the NY Lotto website, in that they say that 40% of revenues go to the jackpot).

Given these odds, how much do you expect to win if you buy a single ticket (good for choosing two six-number combinations)? Well, given the odds of 1 in 22.5 million, you would clearly expect to win absolutely nothing!

But mathematicians don't think this way. Instead, they compute the expectation as the long run average, and by long-run, I mean really really LONG-run (actually infinite-run, but let's not split hairs). To give you some idea of this, you would need to play around 15 million times to have a 50% chance of winning at least one time--this would take 41,000 years or so if you played 1 ticket a day. So, computing the average after a few thousand or even a few million games is likely to get you an average of 0, which is *not* the correct long-run average.

Instead, this expectation is computed by taking the sum of the probabilities of winning multiplied by the amount won. In the case of the Lotto, then you win $0 in (22,499,999/22,500,000) games and $9,000,000 in (1/22,500,000) games. So, the Expected winnings are(22,499,999/22,500,000) *$0 + (1/22,500,000)*$9,000,000 = 40 cents.

So, you pay a dollar, and "expect" to get 40 cents back. This is why some people call the lottery a tax on stupidity. When people say the lottery is a tax on stupidity they are implictly and incorrectly assuming that utility (to throw in an economic term) is based purely on mathematical expectation, and that the utility from $9 million is 9 million times the utility from $1. Yet I doubt that people are playing the lottery based on some mis-guided mathematical expectation calculation. $1 or 40 cents. Who cares? Either way it's barely worth picking up off the ground.

Smart people who play the lottery are valuing 2 things against each other -- $1 versus a miniscule chance of $9 million -- and deciding that the value of $1 to them is less than the value of the chance at the $9 million. Yes, poor people probably value $1 more than average, but they value a chance, even a small one, of forgetting about their financial woes even more.

Let's look at another game that shows the flip-side of this mathematical expectation conundrum. For all you upper-middle class, non-lottery players out there, consider the following: Would you pay your entire net worth for a 1 in 1,000 chance to win $10 billion? If your net worth is less than $10 million, this is a game with positive expectation. For those of us with less than $1 million hanging around the house, the expectation is more than $9 million, but I doubt you'd find any middle-class person willing to play this game.

Why? Because the risk is too great, no matter what the reward. It is widely recognized that people place different values on risk. Risk averse people are willing to lose a small amount of money (or pleasure) to insure they will not lose a large amount of money (or pleasure), even when the mathematical expectation of their transaction is negative. The best example is insurance (Wikipedia's lottery entry points this out). Insurance companies make money not on stupidity but on the fact that people do not want to take large financial risks.

So next time you hear someone say the lottery is a tax on stupidity, tell them about the mathematician who plays, or about the people who turned down a game with an expectation of $9 million.

Saturday, December 1, 2007

Hospital infections-what are the chances?

For those of us with aging parents, visits to the hospital can become somewhat routine, unfortunately. An overnight stay for minor surgery, as my father recently had, isn't particularly worrisome--except for all I've been hearing about hospital infections.

The CDC estimates that about 100,000 people die from 1.7 million infections that they contract at hospitals each year (this is a 2002 estimate, but recent numbers from less authoritative sources are similar or higher). This is an enormous number, and with 35 million people hospitalized annually, this translates to a 5% chance of getting an infection, and about a 1 in 350 chance of both getting an infection and dying from it.

According to the non-profit organization RID, the 100,000 deaths are more than the deaths from breast cancer, AIDS, and car accidents combined. And how do we prevent these infections? Mostly simple cleanliness and sterilization procedures, such as washing hands and washing equipment properly. Procedures we assumed were practiced by hospital staff across the board.

So what's the good news? Several states now are considering or have passed laws that require disclosure of hospital infection rates for each hospital. Of course, some hospitals will have higher rates simply because they perform more acute care and their patients are sicker. However, these bills would finally put the responsibility back on the hospital, and would allow a more informed choices about where to schedule a procedure (when we have such a choice).

Thursday, November 22, 2007

Thanksgiving Brothers

I had some role in coming up with these statistics in the New York Times, though I thought it was more like 1 in 100 than 1 in a billion.

Lost luggage - 1 in 138?

Thanksgiving is the beginning of peak travel season, and what did I read this morning? That 1 in 138 checked bags gets lost.

Now that seems like quite a bit. It implies that someone on your plane is going to be sitting at their in-law's formal Thanksgiving dinner with jeans and a sweatshirt. Maybe it serves the in-laws right, but after a few days, it's not just the in-laws who care.

Yet the statistic above is skewed for a number of reasons. Most "lost" baggage actually gets found, and often quickly. In all my flying and that of my family, we've never had a "lost" bag for more than 24 hours (and only once was it overnight). (ok, they're biased) cites a Dept. of Transportation statistic that says only 1 in 20,000 bags are permanently lost.

I would imagine there are other issues at play, also. Connections are riskier, since it's pretty hard to screw it up on a direct flight, as you watch them put it on the conveyor and tag it, and they only have to get it on one plane. Of course, the time you really want to check vs. carry on is when you are going to spend 3 hours waiting for your connection and shopping at some random airport.

But let's get back to 1 in 138. What does it mean to you? It means if you are a typical air traveller (10 or fewer flights a year, 1 bag checked each time), you will only get a bag delayed about once every 9 years. Even if you took 50 flights a year, you are not likely to ever have one lost forever.

This assumes all flights are the same. There is a clear difference in international flights and at least some differences among carriers. I'd imagine, a flight from, say, budapest, to New York, is more likely to have a lost bag than a flight from DC to New York. When we went from Whenzhou to Gaungzhou in China, they were so concerned with theft, they made us purchase a 10 yuan ($1.25) lock for every suitcase. I guess it did the trick, because everything arrived, and ontime!