Fun With Expected Value
©2020, George J. Irwin. All rights reserved.

Here's something that doesn't come up quite as often as it could in Black Belt Land: the concept of "Expected Value." A simple textbook definition of this is "the arithmetic mean of a random variable X with equiprobable outcomes." We can make this more complicated by introducing outcomes that have different probabilities, but let's limit this discussion to something that does have all outcomes equally probable: A lottery ticket.

Oh, no, it's the "tax on people who don't understand math," as any number of people have said before me. And to me.

But have you ever stopped to consider why that's so much more true than it isn't?

Large jackpot games do carry, well, large jackpots, but they also carry very large chances of winning. In the version of Powerball that is current at this writing, the odds of picking the five main numbers and the Powerball are 292,201,338 to one.

Applying the math, the expected value is the average of one times the jackpot and zero times all of the other possible outcomes—which is that you don't win the jackpot. I'm going to ignore the other prizes, for now. Technically speaking, that average is not zero, but it's got a lot of numbers to the right of the decimal; how many depends on how large the jackpot is. In other words, you're not getting your two dollars back!

In theory, there is a point at which the mathematical expected value, though not the practical one, actually does climb above the two dollar expense of a lottery ticket. That's when the Powerball jackpot rises higher than two times the odds of winning; that is, averaging (1 times jackpot) with (zero times everything else) is more than $2. What's that magic jackpot number? About $584.6 million! Oh, wait, that would have to be after taxes, no? Neither of the two United States multi-state lotteries have ever hit this after-tax number, not even the $1.5 billion Powerball record-smasher of 2019— because it was split among three winners.

But wait, how about the non-jackpot prizes? I'll pick on Mega-Millions this time: the odds of catching the five regular numbers but not the Mega Ball are over 12 million to 1... but the prize is just one million dollars. The expected value is not in your favor here! And the chances of just getting your $2 back are one in 37—doesn't that imply that the prize should be more than a measly two bucks? By way of comparison, it's a 35 to 1 payout in most Roulette games at American casinos, giving a slight edge to the house since there are 38 possible outcomes (1 to 35 plus zero and double zero), but nothing like the odds versus payoff on lottery tickets. If you don't believe me, try playing every possible one of the 25 Mega Ball choices and see what your net return is. Sure, you might luckily also snag a bigger prize, but it's safe to say your chances aren't good.

And if you think that's bad, take a look at the odds tables for Pick-3, Pick-4, scratch-off or pull-tab tickets sometime. Don't expect much value.

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