After originally posting this, powerball changed the odds and payouts. Instead of drawing the group of 5 from a set of 53, they draw from a set of 55. I assumed that this would be simply a reduction in the expected value of a ticket, but they also tweaked the payouts for 2 of the non grand prize winners, and the result was that the expected value of a non grand prize winner actually increased, from 17 cents to 20 cents.

Some of the increase in expected value was driven by the increased odds of ‘winning’ by *not* getting correct numbers in the group of 5. This is a very small effect, however, since the odds only actually increased for the nil case of getting *no* correct numbers.

This tells me that powerball revenue is affected as much by psychology as statistics. The net effect of the changes was to dramatically increase the odds against winning the grand prize, thus driving the average grand prize upward, thus increasing ticket sales. And since the expected value is still very low, increased ticket sales translates directly to increased revenue. In fact, a 4% increase in sales would offset the increased expected value.

Here is the new analysis, followed by the previous analysis.

Five balls are drawn from a drum with balls numbered 1-55. One ball is drawn from a drum with balls numbered 1-42.

The total number of combinations is ~~~~ total = c(5,55) * 42 = 146,107,962 ~~~~

hits | payout | odds | formula | count | payout |
---|---|---|---|---|---|

0x + p | 3 | 69.0 | c(5,50) | 2,118,760 | 6,356,280 |

1x + p | 4 | 126.9 | 5 * c(4,50) | 1,151,500 | 4,606,000 |

2x + p | 7 | 745.4 | c(5,2) * c(3,50) | 196,000 | 1,372,000 |

3x | 7 | 290.9 | c(5,3) * c(2,50) * 41 | 502,250 | 3,515,750 |

3x + p | 100 | 11,927.2 | c(5,3) * c(2,50) * 1 | 12,250 | 1,225,000 |

4x | 100 | 14,254.4 | c(5,4) * c(1,50) * 41 | 10,250 | 1,025,000 |

4x + p | 10000 | 584,431.8 | c(5,4) * c(1,50) * 1 | 250 | 2,500,000 |

5x | 200000 | 3,563,608.8 | c(5,5) * 41 | 41 | 8,200,000 |

5x + p | grand | 146,107,962 | c(5,5) * 1 | 1 | grand |

In the table above, payout is the amount that would be paid out for each type of prize assuming that all combinations were selected. The total payout for all combinations other than the winner is $20,888,592, or just under one sixth the total amount.

The following table shows statistical expected value of a $1 ticket for a range of grand prizes. I have extended the table to reflect the fact that the grand-prize *nominal* amount is over twice as high as the *actual* amount you would receive (before taxes) if you take the one-time payout option.

Prize | Nominal Payout | Expected Value | Actual Payout | Expected Value |
---|---|---|---|---|

30,000,000 | 58,800,030 | 0.40 | 42,900,030 | 0.29 |

60,000,000 | 88,800,030 | 0.61 | 57,000,030 | 0.39 |

90,000,000 | 118,800,030 | 0.81 | 71,100,030 | 0.49 |

120,000,000 | 148,800,030 | 1.02 | 85,200,030 | 0.58 |

150,000,000 | 178,800,030 | 1.22 | 99,300,030 | 0.68 |

249,591,345 | 278,391,375 | 1.91 | 146,107,962 | 1.00 |

0 | 28,800,030 | 0.20 | 28,800,030 | 0.197 |

Ignoring the grand prize, the expected value of a $1 ticket is 19.7 cents. And the odds of winning the grand prize are so vanishingly small that it is reasonable to ignore it in the expected value calculation. Thus, you can expect that you will on average get back less than $10 for each $50 that you spend - not a good value proposition.

The marketing of powerball, and of state-run lotteries in general, seems to me to verge on fraudulent in 2 ways:

- The stated prize value is misleading at best. The nominal payout is paid out over a 30 year period, rendering its net present value far less than the nominal value. The actual present value works out to about 47% of the stated prize amount. This fact should be prominently stated in all marketing material for the lotteries.
- It is unreasonable to expect that consumers will engage in a laborious statistical analysis of the odds, payouts, and expected values of lottery games. Lottery game purveyors should be required to distribute an analysis, similar to the one taken above, to allow consumers to make better informed decisions about lottery purchases. Failing to do so simply exploits the inability of most people to reason effectively about very large or very small numbers.

I’m sure it is very clever to say that “lotteries are a tax on the (mathematically) stupid.” But as a society isn’t it our responsibility to provide the tools for people to make informed choices? In fact, haven’t we accepted that responsibility in nearly every other area of our lives? We insist on truth in advertising for everything from corn chips to cars. But we exempt the lottery from that same requirement.

### Powerball Analysis (pre August 2005)

Five balls are drawn from a drum with balls numbered 1-53. One ball is drawn from a drum with balls numbered 1-42.

The total number of combinations is ~~~~ total = c(5,53) * 42 = 120,526,770 ~~~~

hits | payout | odds | formula | count | payout |
---|---|---|---|---|---|

0x + p | 3 | 70.39 | c(5,48) | 1,712,304 | 5,136,912 |

1x + p | 4 | 123.88 | 5 * c(4,48) | 972,900 | 3,891,600 |

2x + p | 7 | 696.85 | c(5,2) * c(3,48) | 172,960 | 1,210,720 |

3x | 7 | 260.61 | c(5,3) * c(2,48) * 41 | 462,480 | 3,237,360 |

3x + p | 100 | 10685.00 | c(5,3) * c(2,48) * 1 | 11,280 | 1,128,000 |

4x | 100 | 12248.66 | c(5,4) * c(1,48) * 41 | 9,840 | 984,000 |

4x + p | 5000 | 502194.88 | c(5,4) * c(1,48) * 1 | 240 | 1,200,000 |

5x | 100000 | 2939677.32 | c(5,5) * 41 | 41 | 4,100,000 |

5x + p | grand | 120526770.00 | c(5,5) * 1 | 1 | grand |

In the table above, payout is the amount that would be paid out for each type of prize assuming that all combinations were selected. The total payout for all combinations other than the winner is $20,888,592, or just under one sixth the total amount.

The following table shows statistical expected value of a $1 ticket for a range of grand prizes:

Prize | Total Payout | Expected Value |
---|---|---|

30,000,000 | 50,888,592 | 0.42 |

60,000,000 | 80,888,592 | 0.67 |

90,000,000 | 110,888,592 | 0.92 |

120,000,000 | 140,888,592 | 1.17 |

150,000,000 | 170,888,592 | 1.42 |

180,000,000 | 200,888,592 | 1.67 |

Ignoring the grand prize, the expected value of a $1 ticket is 17.3 cents. And the odds of winning the grand prize are so vanishingly small that it is reasonable to ignore it in the expected value calculation. Thus, you can expect that you will on average get back about $10 for each $58 that you spend - not a good value proposition.