Lottery rollover - fewer players, bigger prizes? Bad maths

I was sceptical of a claim on Free Exchange today:
"...if there is no winner the prize is carried over to the following week. A smaller participant pool can then result in infrequent, higher jackpots."
This struck me intuitively as unlikely. So I thought I would work it out.

Probability theory has been unpopular among economists this year - everyone quotes Nicholas Taleb and slowly, loudly explains to us how the financial markets don't behave like a normal distribution after all, and we don't have enough historical data to give us a predictable distribution for the future. Insufferable.

Fortunately lottery draws do obey standard probabilities and we do have enough data (and enough theory) to predict how they will behave. So we can use some standard results.

Assume that a lottery has 10 million participants each paying $1, with a 50% payback. The prize fund in a typical week is $5 million. Let's say the odds of getting the right numbers are 1 in 14 million; then the chances of a rollover are around 49% (see this page on Poisson distributions to work out why, or post a comment if you want an explanation).

So starting from a week with no rollover, there's a 51% chance of a $5 million prize, a 25% chance that it builds up and $10 million is won the next week, 12% of $15 million the week after, 6% of $20 million, 3% of $25 million... 0.2% of $45 million... and so on.

Now imagine the number of players falls by half. 5 million participants, $2.5 million expected prize. Chances of no winner in a given week are now about 70%. So again starting with an empty prize fund, we have a 30% chance of $2.5 million, 21% chance of $5 million, 15% chance of $7.5 million, 10% chance of $10 million, 7% of $12.5 million, 5% of $15 million... 1% of $25 million... 0.07% of $45 million.

The probability of reaching a given size of prize fund never exceeds the probability with a higher number of players. You can tweak the numbers to check this - it's easy to build a model in Excel - or I may revisit this shortly with an analytical rather than numerical demonstration. But I'm afraid the Economist, for once, has made a mistake.


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