My mother passed on this excellent piece from The Week that discusses the psychology of lotteries. It’s well worth reading – it covers all of the ways that lotteries exploit our mental and emotional weaknesses. But the lead paragraph has one of the most confusing explanations of a lottery that I’ve ever seen. For context, the basic idea behind a lottery is to sell n tickets for $Z apiece, and award the winning ticket some amount $X that is much less that n times $Z. One person wins a big cash prize, and you make a tidy profit. That’s the underlying grift behind all the mind games lottery organizers play. But according to The Week (and the underlying Nautilus post) it’s not how the Powerball drawing on May 18th, 2013 worked:
To grasp how unlikely it was for Gloria C. MacKenzie, an 84-year-old Florida widow, to have won the $590 million Powerball lottery in May, Robert Williams, a professor who studies lotteries, offers this scenario: Head down to your local convenience store, slap $2 on the counter, and fill out a six-numbered Powerball ticket. It will take you about 10 seconds. To get your chance of winning down to a coin toss, or 50 percent, you will need to spend 12 hours a day, every day, filling out tickets for the next 55 years. It’s going be expensive. You will have to plunk down your $2 at least 86 million times.
Williams could have simply said the odds of winning the $590 million jackpot were 1 in 175 million. But that wouldn’t register. “People just aren’t able to grasp 1 in 175 million,” Williams says.
If the jackpot is $590 million, and the odds are 1 in 175 million, then each ticket has an expected value of $3.37. At $2 per ticket, you make a profit, on average, of $1.37 for each ticket you buy. Taken at face value, these numbers imply the folks running Powerball (a cabal with the lovely moniker of “Multi-State Lottery Association”, which is for some reason abbreviated MUSL) lost money on that drawing. There are tons of caveats here: to play the eponymous “powerball” costs an extra dollar, and taking the money all at once reduced MacKenzie’s payout to $370.8 million.* But if these details matter, then they should have been clarified. I’m assuming the real issue is an error in the probability of winning, but a lazy Google search was unable to confirm that.
Irrespective of the cause, this is an extremely telling error to make in an article about why we (foolishly) play the lottery. The author himself couldn’t do the math to see that the numbers implied that the expected profit from the example lottery was positive. This actually shows that Robert Williams is on the wrong track! It’s not just that people can’t understand small probabilities – in general, we can’t compute expected values either.