# "Somebody has to win" a lottery. True? Not true?

Before the big jackpot deal last weekend, a co-worker was saying how she really was thinking of buying a ticket because “somebody has to win.”

But my understanding has always been that that isn’t true in a lottery. Aren’t the numbers drawn at random*? I thought that was how the jackpots grew so large, that there was a drawing no one won, so the money stayed in the pot until the next drawing.

Isn’t this the difference between a lottery and a sweepstakes, where in the latter the winning ticket is pulled from all the entries?

• as random as they can, anyway

That is my understanding too. With the resent giant jackpot, a lot of the money was held over from the previous lottery that nobody won. Those previous players and their tickets no longer had a chance of winning, only those who purchased tickets for the new drawing can win. right?

It’s obviously not true that somebody “has” to win a lottery. When people use that term what they actually mean is that people “do” win lotteries, and therefore its possible that they might also.

It is true that the reason the lotto jackpot got so big is that no one won in previous drawings, and the prize rolled over. But statistically, someone will eventually win. The larger the prize grows, the more people buy tickets, and the higher the probability that someone will win. So yes, for all practical purposes, someone has to win; eventually.

Someone will win in finite time with probability one.

I agree with Fear Itself. For any given individual lottery, nobody “has” to win. But for the march 30th drawing, I believe the lottery said that over 95% of all possible combinations had been chosen, so it would have been very hard for someone not to have won.

Yep. I recall seeing an analysis before last week’s mega millions drawing that estimated there was only a 5-6% chance that no one would win. Had no one actually won, the jackpot would have grown close to \$1 billion, which would likely have juiced ticket sales so much that you’d almost (but not quite) be sure to have a winner in the following drawing.

Technically speaking, if people are truly selecting numbers at random (and not - for example - intentionally buying up all the combinations), no one ever “has to win”. But for practical purposes, yes, someone will eventually win.

Thanks, all.

The only way that a lottery could require a winner is if a ticket was sold for every number combination.

For the MegaMillions lottery, how many number combinations are there?

Not quite 200 million.

It’s not true for the specific drawing if the prospective winning number has not been purchased, but it is true from the player’s prospective as an ongoing lottery purchaser. The drawings will continue. There will eventually be a grand winner. Only someone who plays can win. As the pot continues to grow, many if not most players will continue to play or even up the purchases.

From my prospective, I win by not buying any tickets.

This jackpot was very unusual. The number of consecutive non-wins that led to it should only happen once every couple of hundred years according to a statistician I heard on NPR.

New question, then, with a hypothetical scenario because I don’t know the details of these things: “Big Winner Lottery” is drawn the first of every month. If one buys a ticket for the April first drawing and there is no winner on April 1, is that ticket still valid for the May 1 drawing? I thought you had to buy a ticket for each successive drawing.

The point of the original question was not whether someone will eventually win an eventual drawing eventually, but that my co-worker was under the misapprehension that every lottery drawing had a winner, which I have now received confirmation is incorrect, as I told her at the time I was pretty sure it was.

The money rolls over. I don’t know of any lottery where the tickets rollover. You need to buy a new ticket. Buyers consider themselves “in the hunt” so they keep purchasing as a cost of hunting. (You buy a new hunting license every year even if you never see a buck.)

This is utterly and demonstrably false. It is entirely possible (though increasingly unlikely) that there will be no winner for each subsequent drawing.

I want to see this demonstration.

No, he’s correct that for any length of consecutive drawing there remains a chance for zero winners. At some point it approaches the odds seen in thermodynamic miracles but the chance is always there.

For myself, I prefer the odds of my underwear spontaneously transmuting into gold.

Because that would be awesome.

That’s only assuming of course that people are picking their numbers randomly.

In the real world, if the lottery ever got up to some ridiculously large amount of money, in the billions, then you can bet your bottom dollar that some syndicate is going to come up with a scheme to buy all 177 million possible combinations over the span of 3 days, and take home the jackpot.

Thus a winner, in the real world, is guaranteed (eventually).

There are two true claims here:
[ol]
[li]For any fixed number of drawings, there is positive probability that the prize will never be won.[/li][li]If the lottery is played repeatedly, then as long as there’s at least one ticket sold per drawing, there will be a winner in finite time with probability one.[/li][/ol]
The first is easy to see from the properties of the geometric distribution. The second claim requires some slightly sophisticated machinery to prove, but it can be found in the third edition of Durrett’s Probability: Theory and Examples as Theorem 5.2.3.

I think you’re assuming the U.S. polity (and its lotteri(es)) will continue to exist indefinitely. Just the rhetoric in the present election cycle should call that assumption into question.