Lottery Statistics

Written by: Jennifer Thompson

I read an article about some proposed legislation in North Carolina which would restrict some residents from being able to purchase lottery tickets. These restricted residents include those involved in bankruptcy proceedings and those receiving public assistance. I am going to stay far away from the discussion of whether this proposed law is a good idea and sound legislation or a gross overreach of government power. Feel free to carry out that discussion without me. But I will take the opportunity to talk about the statistics of the lottery.

Let’s talk Powerball. They are nice enough to publish the probabilities of the prizes available to their customers. Let’s use the annuity payout number for the jackpot since it is higher and forget about the taxes. It’s only nearly half your winnings for the larger prizes. So for the Powerball drawing on Saturday, January 26, the jackpot is \$130 million. What is your expected return on a \$2 lottery ticket?

The above chart gives the odds and prize amount for each of the possible ways to win regular Powerball as well as Powerplay. (Powerplay costs an additional dollar and increases all prizes except the jackpot.) The prize amount and odds of winning are taken from the Powerball website. The two columns of returns are calculated by taking the prize value and subtracting off the cost of the ticket. This is \$2 for a regular ticket and \$3 for a Powerplay ticket. Then divide that difference by the odds of winning that prize. When we sum the expected return columns, we get the overall expected return for a lottery ticket purchase. The expected value of a regular Powerball ticket for the Saturday drawing with a \$130,000,000 jackpot is \$1.04 (for a \$2 ticket purchase and a loss of \$0.96) and for Powerplay, the expected return is \$1.51 (on a \$3 purchase, a loss of \$1.49). (Note that the return column is rounded to the cent. The sum is performed on the raw data before rounding.)

Expected returns are generalized over the whole population of lottery customers. Any statistician knows that rare events do happen. A 1 in 175,223,510 chance means that with enough players, someone is getting lucky. The census claims the US population is 315,218,311. So if everyone in the US plays 1 ticket, 1.8 people should win jackpots. Wait, you have to be 18 to play. So it’s a little less than that. But thinking of it that way actually makes it seem more likely to me that someone would actually win.

An interesting side note, the folks at Powerball actually advocate buying 35 tickets, each with a different Powerball number so you are guaranteed a win of at least \$4 on your \$70 purchase. Doesn’t sound like smart money to me, even with this method of beating the system. (See their FAQ section. There is no anchor on the page to link to the exact spot, sorry.)