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.