A whole Lotto luck

by Burkard Polster and Marty Ross

The Age, 13 July 2009

It’s always the way. The Maths Masters decide to take a short, well-deserved holiday, and Australia is immediately flooded with numbers and probabilities. We’ve returned just in time to watch the final, receding waves of Lottomania.

Your Maths Masters didn’t participate in this gambling orgy. However, our partners did - as did our friends, relatives, colleagues and possibly some of our pets.

As certified mathematical watchdogs, we don’t object to a small flutter on a lottery; a few dollars is a small price for the dream of easy wealth. True, we were not thrilled that mathematicians had to share commentary space with numerologists and fortunetellers. And we could have done without the sight of snaking queues outside of “lucky” lotto agencies. But, all in all, we had no problem with the lotto craze.

Still, there is some mathematics to be done. The media had a lot of fun comparing the chances of sharing in the Big Prize to being struck by lightning and so forth. So, let’s begin by working out those odds.

The Big Prize is won by correctly choosing seven numbers drawn out of 45. There are then 45 choices for the first number, followed by 44 choices for the second, and so on, down to 39 choices for the seventh number. Multiplying the choices, this gives a total of about 225 billion ways the numbers might be drawn.

However, the order in which the numbers are drawn doesn’t matter. For a given selection of seven numbers, a similar calculation to the above shows that there are about 5000 different orderings. Then, dividing 225 billion by 5000, this gives about 45 million different combinations.

So each lotto entry has one chance in 45 million of sharing in the Big Prize: not great odds. Is there anything we can do to improve matters? Not hugely, but there are a couple of worthwhile strategies.

First of all, if we happen to fluke a prize, we’d prefer not to share the prize money with too many others. So, we should try to choose number combinations that others would avoid. Our ticket above suggests one promising choice – unless we’re tripped up by other mathematicians with the same clever idea.

The other strategy is to wait exactly for the situation that has just occurred, where money has jackpotted from previous lotteries. Obviously, having free extra money added into the pool is a good thing.

Let’s consider the recent lottery, where $60 million from previous lotteries was included in the Big Prize. This contribution raised the total allocated to the Big Prize to about $110 million. Perhaps it would have been worth purchasing a “System 45”, playing every combination and guaranteeing a slice of that huge prize?

To play every combination would have cost $45 million, plus about another $5 million in commission. The question is, how much would we expect to win back?

Let’s first consider the many smaller prizes we would win. About a third of the total money wagered is returned in small prizes. This would return to us about $15 million of our $45 million.

Now, what about the Big Prize? By playing every entry we’re of course guaranteed to win a slice of this, but alas we’ll probably have to share it. In fact, there were about 215 million entries in the recent lottery, each with one chance in 45 million of sharing in the Big Prize. Doing the division, we might guess there to be about four winners, not counting ourselves.

Our huge entry would also have increased the pool for the Big Prize, up to about $125 million. Dividing this amount between the four hypothetical winners and ourselves, this would return about $25 million each.

Now we can sum up. Given a guess of $25 million as our slice of the Big Prize, together with $15 million from all the smaller prizes, that totals to about a $40 million return on our $50 million investment. Dang!

It all looks to be a mighty impressive way to blow $10 million. However, as it happened, only two winners shared in the Big Prize, and we would in fact have ended up about $5 million ahead. So, perhaps next time we’ll give it a go. We’re just looking for a financial partner willing to front us the missing $49.98 million we need to get started.