# Punk Rock Operations Research

## Bulgarian lottery – what are the odds?

Posted by Laura McLay on October 7, 2009

The same six winning lottery numbers turned up in two consecutive drawings in the Bulgaria lottery earlier in the month (1 chance in 5.2 million).  Carl Bialik in the WSJ writes about the odds of this happening.  He notes that “With so many numbers colliding each week, the lottery might be the ideal proving ground for something that statisticians have long recognized: Given enough opportunities, the seemingly impossible becomes plausible.”  He explores several lottery issues in more detail in the Numbers Guy blog.  Statistician David Smith also blogged about the Bulgarian lottery.

Although the lottery is random, the people who play it are not.  I had always intuitively known this, but the picture below illustrates this quite nicely.  Apparently, people making lottery picks based on birthdays, for example, skews the picks toward smaller numbers.

Lottery numbers as chosen by lottery players are far from random

The lotteries are designed such that the expected winnings are negative when accounting for the price of the ticket, since the probability of winning is so low (E[winnings] = P(win)*Jackpot – Ticket Price). When the jackpot grows large enough, the “average” lottery player can come out ahead (although there really is no one at the average – there are a couple of winners who really skew the average). In March 1992, the Virginia lottery almost guaranteed a true winner. It offered a jackpot of \$27M to a single winner whereas it cost \$7.5M to purchase all Choose(44, 6) combinations of possible tickets (by piacking six of 44 numbers). Of course, this strategy could backfire if there were many winners. However, a group of 2500 people accepted this challenge and pooled their resources. They ended up being the single winner, and after a legal struggle, they were awarded the jackpot. The Virginia lottery was subsequently changed to be less lucrative.

Do you play the lottery?

1. ### iamreddavesaid

There is a good article on bias in lottery picks in plus magazine
http://plus.maths.org/issue29/features/haigh/
I looked at the issue here http://liveatthewitchtrials.blogspot.com/search/label/lottery
There are links to various papers on the topic ther as well.

Most lotteries have returns of about 50% and you can boost it to nearer 75% if you pick your numbers cleverly.

2. ### Laurasaid

Thanks for the links!

3. ### Larry (IEOR Tools)said

I don’t play the lottery. I’m a statistician. Enough said.

4. ### Philipsaid

I’m surprised the decline in fraction-of-people-choosing a number as the number size increases (for the Powerball ’96) was so… gradual. I would have expected several, rather sharp, discontinuities.

5. ### Laurasaid

I think that many lottery tickets sold use randomly generated numbers, since not everyone wants to pick their numbers. This may accountfor the gradual decline in fraction-of-people-choosing a number. But I have never played the lottery, so I could be way off base here.

6. ### Philipsaid

Ooh, that makes sense, and I think it’s right, Dr. McLay. I suppose that would both flatten the distribution and provide pressure towards some kind of approximate continuity. Consider my skepticism addressed!

7. ### Aditya Aneysaid

I wonder if winning the jack pot had anything to do with E(winnings).. it seems to me more like, it makes sense to pool your money, if :

jackpot > (total no of combinations)*ticket price

or is that the same as saying E(winnings) > 0 ?

8. ### Aditya Aneysaid

hey! it is the same thing:

jack pot > (44,6)* ticket price
=> (1/(44,6))* jackpot > ticket price

=> P(winning)*jackpot > ticket price > P(losing)* ticket price

=> E(winnings) > 0

sorry for the hassle..!

9. ### Laurasaid

No problem. Thanks for the clarification!