Wikipedia:Reference desk/Archives/Mathematics/2013 December 15

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December 15

Mathematics of lotteries like Canada's Lotto Max

I have a question about "beating the odds" in lottery. Please note this is strictly a mathematical curiosity, and I'm not trying to get rich by gambling. I'm going to be using Canada's Lotto Max lottery as an example, but this idea should apply to most lotteries. In Lotto Max, 45% of the ticket sales is dedicated to prize money. This number is constant in the long term, and in weeks where the jackpot(s) are not won, they are "rolled over" and added to the next week's prize money. The main jackpot is capped, and additional smaller jackpots are created along side it if this goes on for multiple weeks. All of the jackpot money is inevitably won, and the lottery goes back to its standard jackpot.

Beating the system: On a national, long-term average of all gamblers, every gambler will get back 45% of the money he gambled away. The "house", or the provincial lottery corporations and the network of dealers, will keep 55% of the gamblers' money. So let's just say, you were an investor, willing to buy extremely large numbers of tickets as an investment, given a large enough sample size (i.e. you buy millions and millions of tickets over a long period), you can expect to lose 55% of your money.

But this only applies to distributing your "investment" more or less evenly and blindly. The fact is, many weeks have no jackpot winners. In those weeks, the average gambler gets back maybe 5% of their money. (This number can be calculated as the total of all small prizes paid out, as a percentage of the ticket revenue that week. The weeks immediately after that, are statistically, over the long term, certain to pay back more than 45%, to keep the overall average at 45%. So if this "investor" decides to never buy tickets on "standard" jackpot weeks, and always buy his tickets when the jackpots are inflated, he would (again, given an extremely large sample size) statistically expect to be paid back more than 45% of his money, thereby beating the odds that a non-selective gambler has.

This effect can be greatly increased by only buying tickets when the jackpot has not been won for several consecutive weeks, is maxed out, and there are many additional smaller jackpots. For example, this week, as of this writing, the jackpot will be 50 million, plus 50 x 1 million jackpots, for a total of 100 million, compared with the standard 10 million jackpot.

This "selective" buying can clearly improve your odds over the 45% payout, but I'm wondering if this will every push them beyond 100%. What I'm trying to figure out is if it's possible that this "selective" buying can, ignoring chance (which over an extremely large sample size is actually predictable, for example flipping a coin a million times will almost inevitably result in heads being 49-51%) can be used to actually gain a simple predictable edge over not just the other gamblers (earning more than 45%) but over the house (earning more than 100%).

Again, I can't stress the issue of sample size. This is basically considering that you are a billionaire willing to literally buy a billion tickets or more over several years of selective buying. Remember that the odds of winning this lottery are something like 1 in so many million, not billion, so by buying a billion tickets, you are unlikely NOT to win several times. It's just a matter of winning a little more or a little less than the statistical average you expect to win. The more you spend, the less it will vary from that number. In fact, taking this thinking to a further (impossible) level, if one gamble spent not billions, but trillions of dollars, then he would be able to with almost complete certainty achieve a number close tot that "average" (agian, think flipping a coin a million times).

Okay, that's the whole question! Now please give me some solid, foolproof, simple explanation why this WOULDN'T work, because I don't see any billionaires beating the system! Thanks for your input. — Preceding unsigned comment added by 2001:4C28:194:520:5E26:AFF:FEFE:6AF8 (talk) 19:59, 15 December 2013 (UTC)

Note that if a week's jackpot incorporates rollover from the previous week, the formula for the jackpot is A + .45Cn, where A is the rollover, C is the cost of a ticket and n is the number of tickets sold for that week. Meanwhile, the probability of any given ticket winning is 1/n. Thus the expected value of a ticket is ${\frac {A+.45Cn}{n}}$, which decreases as n gets larger. If the expected value of a ticket is more than its cost, a rational gazillionaire will buy tickets; but doing this drives down the value of a ticket. Thus we would expect that the value of a ticket to reach equilibrium at its cost -- if it were above the cost, people would respond by buying more, which would drive the value back down. In fact, people are willing to buy tickets even when their expected value is significantly less than their cost, which just drives the value down even further.

So no, with the payout structure you describe, there's no way to get an expected return greater than your outlay. (But I should mention that occasionally, usually as some sort of promotion, lotteries will use payout structures where it actually is possible to have a greater expected return than cost.)--80.109.80.78 (talk) 20:25, 15 December 2013 (UTC)

In fact, I question your claim "This effect can be greatly increased by only buying tickets when the jackpot has not been won for several consecutive weeks". When the prize pool gets uncharacteristically large, people buy more tickets; I'd be curious to see the data for ticket sales vs. prize pool. I suspect that the expected value of a ticket peaks at a small rollover.--80.109.80.78 (talk) 20:56, 15 December 2013 (UTC)

National Lottery (Ireland)#History of Lotto doesn't say how many tickets were sold in total, only how many were sold to a syndicate, so I don't know their expected winnings but it sounds like they were positive. I don't understand why the lottery tried to stop the syndicate. At first it seems to me the lottery should be happy for anyone buying tickets, but maybe they were worried the syndicate would decrease total sales in future drawings. PrimeHunter (talk) 01:58, 16 December 2013 (UTC)

Yeah, I also don't see the issue (although that page definitely needs rewriting for neutral point of view). The syndicate got lucky that no one else had the same idea, or they probably would have walked away with a loss.--80.109.80.78 (talk) 02:17, 16 December 2013 (UTC)