Can someone help me figure out how the lottery gets its numbers?

[xiphias]

So, the Mega Millions jackpot is up to 540 million dollars, and might be raised to 600 million dollars, or, to put it another way, about what's in Mitt Romney's couch cushions.

So, naturally, I've been doing the calculations to see if it's worth playing.

Here's how the game works.

You've got a panel of numbers up top, from 1 to 56, from which you choose 5. Then you've got a panel of numbers on the bottom, from 1 to 46, from which you choose 1. Match them all, win whatever the jackpot is -- it starts from 12 million dollars, and goes up from there, right now being above half a billion dollars.

It seems to me that the odds of winning are therefore 1 in 56*55*54*53*52*46, which is 1 in 21,085,384,320. But the Mega Millions websites all claim that the odds are 1 in 175,711,536, which is exactly 120 times as good.

What are we doing differently? Where are they getting that 120-times improvement in the odds, and which one of us is right?

Edited to add: In any case, if the numbers the lottery web pages give are right, and I've done my math correctly to include all the non-jackpot prizes, the break-even point at which it's worth playing is when the jackpot hits $155,002,177. Not counting taxes, or the odds of more than one person hitting the same number as you, which would leave you splitting the jackpot.

Edited to add again:: Factoring in the fact that the jackpot amount is an annuity, and the cash value is approximately 58% of that, the 25% federal tax, and a 6.25% state tax, the break-even point is $389 million dollars. Going more decimal places than that is pointless, since the cash value is only an approximate percentage of the published jackpot. That still doesn't factor in the chances of hitting the same number as another person and splitting the pool.

Comments

[plantmom.livejournal.com]

I dunno, but I lold at the image of Mitt Romney's sofa cushions.

[asciikitty.livejournal.com]

I haven't done the math out, but I bet it's because order doesn't matter.

If I choose 37, 3, 42, 15, 21 and then 37, that's the SAME as me choosing 3, 37, 15, 21, 42 and 37. Unless order does matter, in which case, never mind.

[ron_newman]

I think that's right -- which means you have to divide the product by 5*4*3*2*1, which is 120.

[xiphias.livejournal.com]

Order doesn't matter -- indeed, as you are marking five choices from a list of numbers, it's always going to be presented in numerical order. You mark them as you go. But my calculations OUGHT to be for the ones where order doesn't matter.

[xiphias.livejournal.com]

That certainly explains where the 120 comes from, but I still don't get why.

[ron_newman]

Let's make this simpler. Say you are choosing 3 numbers between 1 and 4 inclusive, without regard to order.

Your calculation would give 4*3*2=24, but this is wrong, because it counts all of these as different, when they are actually the same:
(1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1)

You have to divide by 3*2*1 to get the correct answer, which is 4. (The same as choosing 1 number to leave out of your 3.)

[xiphias.livejournal.com]

Okay, I see. Yeah. I was figuring that you pick one from a pool of 56, then pick one from a pool of 55, and so forth -- but the fact is that there are 120 different pools that could result from that process, resulting in the same number. Got it.

[zachkessin.livejournal.com]

Your math only works if you don't count time. Which is to say that the EV is positive, but only if you ignore the fact that on average it will take you millions of years to hit and the upper limit on human lifetime is around 100 years.

[ron_newman]

What we're illustrating here is the difference between permutations (where order is significant) and combinations (where it isn't). This may be helpful.

[xiphias.livejournal.com]

Oh, we went over this junior year of high school; it's just that that was over twenty years ago. The information is still there in my head somewhere; it's just retrieving it is tricky.

[ron_newman]

Yep, it's pretty rusty for me as well; I haven't thought about this stuff in years.

[xiphias.livejournal.com]

Well, I COULD simply buy a million tickets at a pop, had I the cash, which would be another way to increase the chance of payout. Y'know, if I had a million bucks.

[vvalkyri.livejournal.com]

elsejournal, someone noted that if he put his $1 into retirement planning it'd likely be $2, if lucky, years down the line, but if he put his $1 in a ticket for The Big Jackpot he could spend 24 hrs daydreaming on What If, which he found worth the monetary difference.

[chanaleh.livejournal.com]

Wait, so, I missed something. How are you defining "worth playing" and "break-even"? Are you talking about buying 175,711,536 tickets in order to force a "win"?

[xiphias.livejournal.com]

I'm defining it as economists do: expected return times chance of return greater than one.

[mrmorse.livejournal.com]

The math term for this is "expected value". You multiply the $540 million payout by the 1/175,711,536 chance of winning, and add that to $0 times the 175,711,535/175,711,536 chance of not winning. The resulting number is the expected value, or the long run average amount you would expect to win per game, if you played a lot of games. To a first approximation, if the expected outcome is greater than the cost of playing, it's to your financial advantage to play.

In lotteries where you can win smaller sums for matching only some of the numbers, you need to add in those probabilities too. So if there was a lottery where you had one chance in 100 of winning $5, and 1 chance in 10 of winning $1, and you lose the rest of the time, the expected value is 5 x 1/100 + 1 x 1/10 + 0 x 89/100.

For the Megamillions, you also have to factor in the probability of splitting the jackpot, which is dependent on the number of tickets sold, and generally makes the whole thing messier and less likely to come out ahead.

The other thing is that the second most important number is the standard deviation, which is really huge in this case. This means you have to play a lot to ensure your outcome is close to the expected value, and leads to the conclusion that even if the payout is big enough, there are much faster and less risky ways to invest your money.

[(Anonymous)]

Can't we just call the lottery a tax on the statistically challenged?

[mrmorse.livejournal.com]

Brad Plumer reports that the expected value of one ticket is 63 cents. As usual, the only way to win is not to play.

[asciikitty.livejournal.com]

whereas I was raised by math teachers, and think about this stuff for breakfast. I'm pretty sure this is a sign of something terrible in me.

[xiphias.livejournal.com]

Of course you can. My question is whether you'd be right to do so. There comes a point where it is statistically beneficial to play. That's what this calculation is about -- figuring out what that point is, and whether we're there yet.

[xiphias.livejournal.com]

My calculation comes out to an expected value of $1.57, which would make it well worth playing, but I'm not factoring in the odds of splitting the prize.

[xiphias.livejournal.com]

The primary distinction is that he DOES take into account an estimate of the odds of splitting the prize two, three, four, or even FIVE ways, which, at the number of tickets sold, is actually pretty likely.

[thnidu.livejournal.com]

Oh, I dunno. I'm a research linguist, and my dear wife was an English Lit major and a librarian. She wrote up the following about our kids (I knew I'd written this up, but I had to find it here):

SON got into trouble in AP English because he didn't want to work out of the intro to poetry book. I had to point out to the teacher that he and I had already hashed out assonance and alliteration as I drove him to nursery school one day. Not to mention that he had written a pamphlet on the subject himself for Academic Decathlon. His teacher got his permission to use it for her regular track 9th graders, since there wasn't anything available that explained it as clearly.

Not to mention hearing DAUGHTER on the phone in high school, saying impatiently to a friend, "Of course I know what the Great Vowel Shift is. Everyone does."

I have not yet taken a poll of 15-year-olds to verify this.

[ragman-jack.livejournal.com]

Isn't there also a statistically probable bunch of numbers people are likely to play?