September 22, 2011

# Harder to win big in the lottery

The ‘Big Wednesday’ lottery has moved from 6 balls out of 45 to 6 balls out of 50, which reduces even further the chance of getting the division 1 prize.  To win division 1, you need 6 balls correct out of 6, plus a correct coin toss.  There are 8,145,060 ways to choose 6 balls out of 45, and about twice as many ways, 15,890,700, to choose 6 balls out of 50.  Adding in the coin toss halves the chance of winning: the chance of winning per ‘line’ used to be 1 in 16,290,120 and is now 1 in 31,781,400.   For a minimum \$4 ticket, which gives 4 ‘lines’, the chance of a division 1 prize was 1 in 4,072,530 and is now 1 in 7,945,350.

One back-of-the-envelope way to get roughly the correct impact of the change is to note that the chance of matching a given ball has gone down about 10%: from 1/45 to 1/50.  Multiplying 90% by itself six times says that the chance of winning is 53% of what it was, a very good approximation to the actual ratio, which is 51%.  The Dominion Post had the correct change, but the computations they report seem to have the effect of the coin toss backwards, so all their probabilities are overly optimistic by a factor of four.

Of course, the other way to look at it is that your chance of not winning division 1 with a \$4 ticket has gone from 99.99998% to 99.99999%. Hardly seems worth mentioning.

Here’s how I did the computations (in R), for anyone who might be interested or who might disagree:

`Number of combinations of six balls`
```> choose(45,6)
[1] 8145060
> choose(50,6)
[1] 15890700```
```Double the number of combinations for the coin toss
> choose(45,6)*2
[1] 16290120
> choose(50,6)*2
[1] 31781400```
```Divide by 4, for the 4 lines giving four chances
> choose(45,6)*2/4
[1] 4072530
> choose(50,6)*2/4
[1] 7945350```
```How much harder has it got: exactly
> (choose(45,6)*2/4)/(choose(50,6)*2/4)
[1] 0.5125677```
```Back-of-the envelope calculation
> 0.9^6
[1] 0.531441```
```Turn them into probabilities
> 1-1/(choose(45,6)*2/4)
[1] 0.9999998
> 1-1/(choose(50,6)*2/4)
[1] 0.9999999```

Thomas Lumley (@tslumley) is Professor of Biostatistics at the University of Auckland. His research interests include semiparametric models, survey sampling, statistical computing, foundations of statistics, and whatever methodological problems his medical collaborators come up with. He also blogs at Biased and Inefficient See all posts by Thomas Lumley »

• Stephen Cope

There is a giant Big Wednesday advertisement on a bus stop I use. I noticed the prizes seemed larger than the previous advertisement I had seen a few years ago. This must be the reason why.

I have heard it argued that once the prize is a certain value it is a better proposition to purchase a ticket than not to. What are your thoughts on this?

• Thomas Lumley

There certainly can be a positive expected return from betting if the jackpot is sufficiently large. There are two reasons why this is rarer than some people think.

1. Other people will also be betting: if you have to share the top prize, the payback is smaller. Since the odds and jackpot are public knowledge, there should never be a big expected return — if there was, everyone would be doing it.

2. Nonlinear utility of money: 10 million dollars is not worth anywhere near a million times as much as ten dollars for most people.

I don’t know whether the ‘must win’ jackpots in the NZ lottery are large enough to give positive expected return.