Some sobering numbers about the Powerball lottery (other lotteries are in the same order of magnitude) from Christopher J. Mecklin and Robert G. Donnelly of Murray State University. ['Powerball, Expected Value, and the Law of (very) Large Numbers' in the Journal of Statistics Education Volume 13, Number 2 (2005)]
...to be at least 95% confident of winning money on the Powerball, even if we are disciplined enough to only play when the expected value is positive and fortunate enough to be the unique winner when we hit the jackpot, we will have to play hundreds of billions of times. This would require hundreds of billions of dollars (which we don’t have) and hundreds of billions of opportunities to play Powerball when the jackpot is high. Even if we buy hundreds or thousands or even millions of tickets when the expectation is positive, we will probably die long before we are ahead. If we lowered the confidence level to 50% or even 10%, we would still expect to need to play 10 to 20 billion times to realize a profit.
Even if we did have hundreds of billions of dollars at our disposal to play Powerball, we probably wouldn’t want to. Risking $200 billion for the chance to win a prize of even $200 million would be proportional to a person with a yearly income of $50,000 risking that entire salary on a game of chance for the opportunity to win a $50 prize.
They go on to show that your opportunities to play with a positive expected value is limited.
Table 5. Minimum Size of Powerball Jackpot Needed for a Positive Expected Value.
| Number of Players | Minimum Jackpot $ |
|---|---|
| 10,000,000 | $293,057,107 |
| 20,000,000 | $305,207,483 |
| 30,000,000 | $317,679,592 |
| 40,000,000 | $330,472,332 |
| 50,000,000 | $343,584,163 |
| 60,000,000 | $357,013,121 |
| 70,000,000 | $370,756,823 |
| 80,000,000 | $384,812,481 |
| 90,000,000 | $399,176,916 |
| 100,000,000 | $413,846,569 |
It is typical for the Powerball lottery to have about 10 to 20 million players for the smaller jackpots (i.e. drawings held after wins) but at least 50 million players when the jackpot has been built up after several successive drawings without a jackpot winner.
The math used in the paper is facinating. It's definitely worth a look.
