## The best place to buy PowerBall Lotto tickets?

by

As noted in a press release from the Oklahoma Lottery, the convenience store “Station 2” in Watts, Oklahoma, has sold two winning PowerBall tickets within the past two weeks.  “Winning” in that they both were redeemable for hundreds of thousands of dollars; neither one won the whole enchilada.

There are probably fewer than 2500 Lotto retailers in the state of Oklahoma.  [The OK lottery printed 2500 copies of the Nov 2007 Winning Ways Retailer Newsletter, so I’m taking that as an upper bound.]   A brief perusal of the OK lottery’s list of recent winners shows that a PowerBall prize of $10,000 or more is won in the state once every other day. Let’s assume there are 2500 Lotto retailers, with notably large prizes being won every other day, and assume that each retailer is equally likely to sell a big-winning-ticket [this is probably not the case, as not all retailers will sell the same number of tickets each day]. The probability that two or more$10K+ prizes would be sold by the same retailer within the last six months would be $1 - \prod_{k=0}^{89} \frac{2500 - k}{2500} \approx 0.80$

If we restrict ourselves to the really big prizes ($100,000 or more), we find that there have been 10 of those in Oklahoma in the past six months. The probability that two or more of those ten would have been sold by the same retailer would be $1 - \prod_{k=0}^{9} \frac{2500 - k}{2500} \approx 0.017$, which certainly makes this a surprisingly unusual event. Or is it? There are 31 members of the Multi-State Lottery Association; as a quick number hack, assuming for the sake of convenience that OK’s numbers are representative of all 31 members, the probability that no two$100K+ tickets sold nationwide in the last six months would have come from the same retailer would be  $\left(\prod_{k=0}^{9} \frac{2500 - k}{2500} \right)^{31} \approx 0.57$, so we’d expect the likelihood that this sort of coincidence would occur somewhere to be roughly 43%.

All in all, pretty good odds.