January 19, 2014

How to beat Lotto

That is, how to gamble in a way that over a course of a year, gives you a higher chance at a larger payout than playing NZ Lotto each week and hoping for Division 1. We all know you can’t “beat Lotto” in the usual sense of improving your odds of winning.

In the ordinary Saturday Lotto, you pick 6 numbers out of 40, and if all 6 are correct (which they aren’t) you win $1 million. The chance of winning is 1 in 3838380 per ‘line’. Suppose you play the minimum of 4 lines, for $6, each week for a year. The chance of winning in a year is one in 18453.75. That is, on average you’d expect to win once in every 18453 years and 9 months.

Alternatively, suppose you save up the $6 per week, and then at the end of the year go to a casino and play roulette.  Put it all on a single number.  If you win, put it all on a single number again, and then if you win,  put it all on a ‘double street’ of six numbers.  Your chance of winning (in double-zero roulette) is 1 in 9145.33, and if you win you will make $2426112.

So, you get twice the chance of winning as you would have for Lotto division 1, and more than twice the payout. The expected return is 85%, much better than the 56% that NZ Lotteries returns (averaged over all its games, annual report).  Does that mean it’s a good idea? No. Not even slightly.  You have a 37 in 38 chance of turning up with $300 and losing it in a few minutes. If you don’t, you have a 37 in 38 chance of losing $7500 in the next few minutes, and if you don’t, you have about an 85% chance of losing more than quarter of a million dollars.   This strategy makes your losses obvious, which makes gambling no fun. And you still only win once every 91 centuries.

Enjoyable gambling, including Lotto, is based on making your losses less obvious by masking them with small wins and stretching them out over time. Of course, that’s also what makes gambling, including Lotto, potentially addictive.


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 »


  • avatar
    Nick Iversen

    The Stuff article seems poorly written.

    “millionares” should be millionaires.

    “the odds increase to 1 in 38 million” should be odds decrease.

    “271 times out of more than 1300 draws” is a bit vague – why say “more than”? Why not put in the exact number. I think it must also include bonus numbers cos using 6/40 and 1300 gives 271 as 6 standard deviations from the expected number.

    “further reducing your chances of winning” contradicts Lumley’s statement that “you can’t affect your chances of winning.”

    “Lumley concedes…” – I bet Lumley made no such concession. It’s not a concession to say that shops with more customers have more winning customers.

    Actually I feel a bit churlish pointing out those defects. I found it to be quite an interesting read with come interesting sources and examples quoted. And the ultimate advice – gamble for the enjoyment rather than the expectation of winning – is pretty good. I was set on the wrong track by the spelling mistake in the first sentence!

    3 years ago

  • avatar
    Thomas Lumley

    Arguably, “odds increase” is a valid variant – I don’t know what the style book says, but there is a respectable body of opinion that says 10-1 is higher odds than 2-1.

    Otherwise, yes. It’s a very welcome change from the utter garbage that often gets written about the lotto. As is last year’s story by David Fisher linked in the last sentence.

    3 years ago

    • avatar
      Nick Iversen

      We can put “odds increase” in the same category as “ten time smaller” I guess.

      3 years ago