# Statistics question

OK, from this web site I read the following text:
Once again, this is false. If you buy one lottery ticket in the Powerball lottery, your chance of winning is 1 in 120,526,770. If you buy two tickets, your odds are 2 in 120,526,770. You have not doubled your chance at winning the lottery; you’ve just decreased the odds against you by an infinitesimal amount.

What’s up with this? 2/120526770 is double 1/120526770! I know the odds of winning don’t increase dramatically, but that’s no reason to say it isn’t double. I can’t convince a coworker that this website is wrong. He simply believes that whatever he reads online is true. I’m hoping he’ll believe me if he sees some reponses online here.

Looks wrong to me.

But remember, twice zero is still zero…

Edit to remove misunderstanding.

Yes, you are right. Your friend must have fallen asleep during the unit on fractions. But keep in mind: the lottery is a tax on people who are bad at math to begin with, so don’t waste a lot of time trying to educate him.

Actually, the answer depends on how the tickets are chosen. If they are chosen at the point of purchase (by the machine) then your chances are not quite doubled because of the possibility of duplication. It would be possible to purchase 121,000,000 tickets and still not win because of this. However, if you chose the numbers yourself the chances do indeed double, as long as you are careful to choose unique combinations.

OMG, this conjures up the wonderful vision of someone buying a handful of lottery tickets with the same number on each one. I think I just wet myself.

Either there’s a typo in there, or the author was confused regarding odds and probability.

The site contradicted itself. Going from 1 to 2 is doubling, by definition. Of *course * it’s wrong. As noted above, that doesn’t mean your chances went up by a significant amount.

If you scroll down to the comments, the author is corrected on this error, but says he has no way to edit the article.

:rolleyes: You need a calculator to verify that 2/120526770 is double 1/120526770?

CanTak3’s coworker should pay him a considerable sum of money.

If you are just picking random lotto ticket numbers then he is wrong - buying two tickets does not double your chances of winning. Here’s the math behind it:

Let’s call the chance that ticket #1 wins p1, and the chance that ticket #2 wins p2.

Let’s also note that those events are statistically independent (whether or not ticket #1 wins has no effect on whether ticket #2 wins) and they are not mutually exclusive (the range of possibilities includes exactly one ticket winning, both tickets winning, or neither ticket winning).

Given that, the probability that AT LEAST one ticket wins is not p1+p2, but rather p1+p2-(p1*p2). This is a basic formula from probability and statistics, the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection (that is, the probability of both events occurring together).

If you just add p1 and p2, you are counting twice that tiny chance that both tickets happen to be winners. You can’t do that. Imagine the lottery was a single digit from 0-9, if you bought 11 tickets at random would you have a 110% chance of winning? Of course not.

So let’s suppose p1=p2=(10e-8, or 1/100,000,000).

The probability of winning if you buy two tickets = (10e-8)+(10e-8)-(10e-810e-8), which is 210e-8 - 10e-16.

Not exactly twice the odds of a single ticket, although in this case because p1 and p2 are so small, it’s really close.

Now, if you intentionally pick the numbers on your lotto tickets so that no ticket is a duplicate of any other ticket then buying two tickets does in fact exactly double your chances of winning, since you are specifically eliminating the chance of both tickets winning at the same time; the intersection has no chance of taking place and that little subtractive term drops out.

Gosh, no need to go into such depth on it; that’s probably more obfuscatory than clarifying. Everyone agrees: your chances double if you pick two differently numbered tickets and don’t change at all if you pick two identically numbered tickets (well, everyone agrees except the linked author, originally). See post #4. But I think it’s fair to say, when one talks about picking two tickets doubling chances or not, one is implicitly assuming that this refers to picking two distinct tickets.

Hey, this is the Straight Dope. If I’m gonna be nitpicky with the math, this is the place to do it

Heh, fair enough.