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Heavy Tails in Probability Distributions


There is a joke about State Lotteries that says: "The Lottery is a TAX on people who are bad at math". Sadly, this is true. It is also a regressive tax because it takes a bigger portion of the income of poor people who typically are the least educated and therefore the least able to determine just how bad a financial opportunity the lottery is.

The math is really not all that difficult. It requires the understanding of a simple concept: The expectation of an uncertain outcome. Here is an example. Suppose you find yourself in a situation where there are three possible outcomes, each with a different reward. Suppose a benevolent and omniscient dictator knows and tells you in advance the exact probability of each outcome and gives you the opportunity to repeat your choice many, many times. If you take each reward and multiply it by each probability then add up the results you have what is known as the mathematical expectation. Putting some numbers to our example we have:

Expected Value =

The payoffs of $10, $20, and $50 times their respective probabilities of 60%, 30% and 10% sum to an expectation $17, based on a large number of independent situations such as the one described above.

Now suppose there is a fee to play such a game. How much, given that you can play it repeatedly as often as you like while the rules remain the same, are you willing to pay? This little thought experiment is at the heart of the lottery joke. Ignoring the blizzard of possible ways actual State Lotteries allow you to win, let's just take the one that makes headlines. Suppose the jackpot is $10,000,000. The lottery people announce the number of tickets sold and the odds of winning. Suppose the odds are one in 24,000,000, a decimal fraction of .00000004166667. Multiply the payoff times the probability and you have just less than 42 cents. What is the price of a lottery ticket?

The simple conclusion is that people who buy lottery tickets are overpaying for the opportunity. This is the result of failing to understand the notion of an expectation. There is even a germ of common sense in it: If someone offers you the chance to win a dollar based on the flip of a fair coin, DON'T PAY MORE THAN 50 CENTS FOR THE CHANCE!!!