## Sunday, January 10, 2016

### Powerball Odds

So there is a lot of Powerball talk, and I am again intrigued by the possibility of a positive expected value for a lottery.  If you remember I wrote a post about Megamillions a few years ago, when the jackpot grew to $640 million. The result, after taking into account both taxes and multiple winners, was that it still wasn't worth playing. I decided to quickly do the same thing for this latest Powerball jackpot. In a separate post, I made a model to predict the ticket sales based on jackpot amount. This was quite a bit of work, so I'm not doing that here. I'm just going to use the data for the latest drawing (Jan 9th, 2016) to see if that one drawing had a positive expected value in retrospect. To begin, the expected value of the non jackpot prizes is$0.32, or about $0.28 if you include tax on the$50,000 and $1,000,000 prizes. Note here, I'm ignoring the "Power Play" thing, which increases the payout for the nonjackpot prizes, but costs an extra$1.

Now we need to look at the jackpot.  The estimated upfront payout was $587,662,740. After taxes that's$340,844,389.  The odds for Powerball recently got way harder, which is why this jackpot is so large.  This drawing had 440,321,172 tickets sold.  Doing the Poisson distribution we see the probability of various numbers of winners based on the odds and tickets sold:

 0 22.1594% 1 33.3922% 2 25.1595% 3 12.6377% 4 4.7610% 5 1.4349% 6 0.3604% 7 0.0776% 8 0.0146% 9 0.0024% 10 0.0004% 11 0.0001% 12 0.0000% 13 0.0000% 14 0.0000% 15 0.0000%

The odds for no winners (which is what happened) was 22%.  This is due to the much lower odds of a jackpot in general.

Adding up the adjusted jackpot shares gives a total of $176,335,787. This could be thought of as the true jackpot value after adjusting for taxes and the possibility of splitting the jackpot. Multiplying by the odds of winning gives an expected value for the jackpot of$0.6035.  Adding the nonjackpot expected value gives a total Powerball expected value of $0.88. The ticket costs$2, so it's not even half the value.

Some thoughts:  The odds were just increased in Oct 2015, and this is the first huge jackpot after that change.  More will follow, so it is possible that the expected value will improve as people get over the hype of a billion dollar jackpot.  However, it is worth noting that even if the jackpot doubled, and ticket sales remained constant the expected value would still be below the ticket price.

I haven't gone into details for the math here.  The calculations and commentaries are the same as the Megamillions post.  So please see that for more discussion.