Don't Just Look at the Odds: The Math Says You Should Buy a Mega Millions Ticket

[brooklynite]

Slate:

The Mega Millions jackpot has reached a staggering $1.6 billion, meaning Tuesday’s drawing will be the largest in U.S. history. The odds haven’t changed, with your chances of winning at 1 in 302 million, and so the value of a single $2 ticket has only increased.

As we wrote when the jackpot was a mere $970 million last week, you can do the math by multiplying the size of the winnings (very large) by the probability of you winning (very low) to get the value of your ticket. For this new $1.6 billion jackpot, ignoring tax implications, the value of a ticket comes out to be $5.53—a very good deal.

We wrote last week that the complications of tax brackets and the variability of state taxes meant that you were probably making a mathematically smart bet by buying a ticket. The conclusion was that, to be absolutely certain the expected value of your ticket was worth more than its cost, you’d have to do a bit of research into your own state’s tax and lottery rules. Well, that’s no longer true. There’s no way to slice the numbers where you don’t come out with a good deal.

Here is how we (with the help of some experts) did the math:

The value of a ticket, as laid out by Jordan Ellenberg, can be found by multiplying the probability of getting a certain prize by the value of the prize itself, repeating for all potential prizes, and then adding your results. If we consider just the jackpot, we multiply the incredibly large prize ($1.6 billion) by the incredibly small odds of winning (1 in 302 million) and get an expected value of $5.29—definitely a lot more than the $2 cost of a ticket.

[exboyfil]

1. Math doesn't entirely work that way

because you have to split the pot with any other winning tickets.

The prize is actually $800M because you have to look at the cash payout value instead of the 20 year annuity (or whatever it is).

Still I might make a stop to purchase a single ticket. Rarely do the house odds favor you so. I already spend $4/week on a pooled Powerball (used to be $2 but the increased cost of the ticket increased how much I spent each week).

[d_r]

2. the odds are still the same

"don't just look at the odds" doesn't really work. It doesn't matter what the prize amount is, it matters what the odds are. Any week, you have 1 in 302 million chance of winning something and a 1 in (302million - 1) chance of not winning. There is no value in not winning, it is 0 (along with this article, I am only talking here about the grand prize). Zero is zero, no matter what the grand prize is.

Now, it could be argued that your odds of winning from buying one ticket are much, much better than from buying none (1 in 302 million is a lot better than none at all). But I don't get the idea of valuing a "ticket" based off the grand prize amount - it is still all (tiny chance) or nothing (vastly larger chance). It isn't like you are going to get $5 back from your ticket.

[LakeSuperiorView]

15. Couple problems with your explanation..

Your odds of not winning are MUCH higher than 1 in (302million - 1).

"There is no value in not winning, it is 0 (along with this article, I am only talking here about the grand prize). Zero is zero, no matter what the grand prize is. "

The value in not winning is -$2, i.e. it costs you to not win, assuming you buy a single ticket.

[d_r]

17. yeah that's true

your odds or not winning are not 1 in (302 million -1) they are (302 million-1) out of 302 million. I got type tied.

[WillowTree]

3. Bottom line: The only thing we know about the ultimate winner is that they had a ticket.

And all ya need is one.

[Bucky]

6. The point is, it's more lucrative to stand on a corner shaking a tin can

[WillowTree]

7. Not for the person or persons who have a winning ticket.

I never buy more than one because one is all that's needed and if I'm not meant to have it, it won't matter how many I get.

It's just fun for some of us to blow $2 and dream a little once in awhile when the jackpot is huge. Try not to be a buzz kill.

[Bucky]

10. I won't stop you from paying the optional tax.

And I apologize for letting facts out to kill your buzz

[Charlotte Little]

73. It is not a fact that...

..."it's more lucrative to stand on a corner shaking a tin can."

You have no idea how much, if even any amount, someone makes standing on a corner begging for money. So no, it's not a fact. It's just conjecture and one tinged with hyperbole.

But you know what? YOU may make more money standing on the corner shaking a tin can. Maybe you're good at begging.

But YOU (or anyone) definitely won't win the lottery if you don't buy a ticket - and that is a fact.

[unblock]

4. factor in taxes and the likelihood that you'll have to split it with at least one other person

and it's not mathematically worth it.

and if you want to get into the diminishing value of dollars in terms of personal utility, then it's definitely not worth it, but that's another story.


but then, lotteries are never about being mathematically worth it....

[LastLiberal in PalmSprings]

8. "...the likelihood that you've have to split it with at least one other person"

"Honey, I just won the lottery. Start packing!"

"Great! Where are we going?"

"We aren't going anywhere. I want you to move out of the house."

[whathehell]

60. The possibility, however slight, of a Billion dollar win isn't "worth" 2 bucks?

Unless you are in very short supply of said bucks, I would have to disagree.

[unblock]

62. The virtual guarantee of losing 2 bucks isn't appealing to me

But knock yourself out if you think otherwise.

[whathehell]

67. Lol

Right.

[liberal N proud]

5. Double your chances, buy 2

I had 76 chances on last weeks drawing, nothing!

[d_r]

13. It doesn't double your chances though

not really if you think about it. Each ticket is 1 in 302 million. It starts over for each ticket.

[qazplm135]

14. if they are different numbers it absolultely does double your chances

from 1 in 302 million to 2 in 302 million.

We know this because if you bought up half of the available number combinations, you'd have a 50% chance and if you bought them all you'd have a 100% chance. Even though EACH individual ticket would individually only have a 1 in 302 million chance.

[d_r]

16. I don't think it goes

to 2 in 302 million. I think that they are both 1 in 302 million. I think the odds reset each time.

Try it with flipping a coin. Each time it is 50-50 heads or tails. It doesn't matter what you flipped before, each time it is 50-50.

Say you flipped a head the first time. The odds of flipping a head the second time isn't one in four (two heads in a row), it is 1 out of 2. If you flipped two heads in a row, the odds of the third time being heads is 50-50, not 1 out of 8. That's how I think it works.

So if you buy one ticket, your odds go way up over buying no tickets, but buying two doesn't make your odds double from buying one, it makes it go up very little. The way I see it.

[Salviati]

18. No, because it's not repicking the numbers after each loss.

If each lottery ticket generated a unique draw, then it would work the way you suggested, but given that there's one draw, buying 2 tickets does double your chances.

Probability is trickier than it seems. Who knew that counting things could be so hard. I had a probability class in college and it was definitely the lowest grade I got in any math class.

[qazplm135]

19. Let's make this easier

let's say I had a lottery with only ten numbers. 1 through 10.

If you picked 7. You'd have a 10% chance of winning right?

Now what if you picked 7 and 3? Each ticket would individually have a 10% chance of being the winner, but you have now covered 20% of the available numbers...thus, your chances would have doubled, from 10% to 20% of winning.

Same thing here, just immensely more numbers.

Your chances have in fact "doubled" from 1 in 302 million to 2 in 302 million, but that doesn't mean much because double of almost zero is still almost zero. Folks get hung up on the idea of double usually being a significant thing. In my example with ten numbers it is, in my example with the lottery, it is not.

[d_r]

21. I understand what both of you are saying

but I am still not sure about it. I'm really not.

[qazplm135]

22. which part are you unsure of?

Let's make it even easier.

I have a lottery of 2 numbers, 1 and 2.

You have a 50/50 shot of guessing right, correct?

What happens if I say you can pick both 1 and 2.

What happens to your odds? They double. Right? From 50 to 100 percent.

1/2 to 2/2. 1/10 to 2/10. or 1/302 million to 2/302 million.
All "doubling" is is multiplying the numerator by 2. It does not matter what the denominator is.
The only thing the denominator does is tell you how significant said doubling is. From completely dispositive in the first, to highly significant (but still somewhat unlikely) in the second, to almost completely meaningless in the third.

Put another way. If you have one apple, and I give you another apple, you've doubled the number of apples you have...regardless of how many apples there are in the world.

[d_r]

25. I think that every ticket

has 1 out of 302 million odds. I know there is only one draw, and so I get the idea that it seems like the odds wouldn't be unique for each ticket but I think really they are. I know that if you have a limited set it will seem like the odds are cumulative, but I don't think they really are. We could do a monte carlo study or something. I think it is a trick of logic. That's just how I see it, and I know that reasonable and very intelligent people see it this way but I don't agree. I will tell you it took a lot of time for my brain to flip from seeing it your way to seeing it mine, but now that it has, it has.

[qazplm135]

27. two ways to break out of that thinking

First, forget about the odds. You have a pool of numbers (or apples).

Each ticket is one of those pool of numbers (or apples).
Each unique set of numbers (apple) you pick is one out of the pool.
Every time you pick ANOTHER unique set of numbers (apple) out of the pool, you are increasing the number of unique sets (apples) you have in the pool.

One apple is less than two apples. Two apples is more than one apple.

If that doesn't work, then again think of it like a jar of jelly beans. You have to guess how many jelly beans are in the jar.
If you guess one time, your odds are pretty low. If you guess a second time (AND you pick a different number) your odds double. The more guesses you get (and assuming you guess a different number each time), the greater your odds of landing on the right number are...even though each individual guess you make has whatever the initial odds are.

[kcr]

38. It took time for your brain to flip to thinking two is not greater than one?

One would hope it wouldn't get there at all. There's no trick of logic, here. You only need one of your picks to win, so the more picks you have, the more chances you have to win. It would be prohibitively expensive but if you picked enough tickets to cover all possible combinations, you'd be guaranteed a win. So, as long as you're picking different combinations for each pick, you increase your odds because you have more possible combos in hand.

[d_r]

43. I'm sorry I disagree

I think that each ticket has a 1 in 320 million chance. Not that two tickets have a two in 320 million chance.

[qazplm135]

55. You're just plain wrong

Indisputably wrong.

[mr_lebowski]

58. Yup ... Sorry D_R ... you are wrong on this one ...

Probability mathematics is very, very well understood.

1 set of numbers is 1/302M, and 2 sets of numbers (now, I'm not certain how many must be different, but it's at least 1, I know that), improves your odds to 2/302M.

[d_r]

66. Ok

That's fine.

[d_r]

65. All right

[fishwax]

39. yes, every ticket has 1 in 302m odds. So if you have two tickets,

Then you've got 2 in 302m odds. The number of possible outcomes is exactly the same, but now you win on two of those outcomes. Buy three tickets and you triple your odds to 3 in 302 million.

[mr_lebowski]

59. This is correct, the only thing I might question is ... how many of the numbers on ticket 2 ...

Must be DIFFERENT from Ticket 1 ... in order to provide this doubling of the odds? Is it just 1 different? 2? All 7? I suspect that there's some 'rule' involved in terms of difference between the tickets in order to actually double one's odds' by buying 2, given that we know that buying 2 identical sets of 7 numbers does NOT increase one's odds (though it would increase one's reward, should you have to split your winnings) ...

[Hav]

64. Just one number being different would be enough

It doesn't matter whether your second ticket varies in one or all numbers, in both cases you have bets on two outcomes out of the 300 million possibilities.

[fishwax]

69. only one number has to be different for the big jackpot (BUT ...)

Because there are a set number of possible combinations and changing one number gives you a new combination that you could win with.

HOWEVER, having all the numbers be different would increase your chances on the sliding prizes. As I understand it, you get some smaller prize for getting five numbers right, four numbers right, etc. The more numbers you change, the more new 3-, 4-, and 5-number combinations you create. So, overall, it makes sense to play entirely new numbers in general. But for the big jackpot that everyone is talking about changing one number is enough to double your (still abysmally-long) odds.

[Dave Starsky]

77. You CAN do a Monte Carlo study on this.

It's fairly easy to write a Python script that will run this scenario as many times as you want, to see how it shakes out over many millions of iterations.

When you do that, you see that your probability of winning increases by an additional 1/(astronomical number) with every ticket you buy for that same drawing.

I had to write a Python script to solve the Monty Hall Three Door problem once, as I just couldn't believe it. (If you're not familiar, it's another crazy counterintuitive problem in statistics. I'll let you Google it.) Sure enough, the results defied what I "knew" to be true.

[lindysalsagal]

47. 2 peas to a starving man are as irrelevant as 1 pea.

Yer still starving.

[customerserviceguy]

79. Yes, I feel better

with two snowballs' chances in hell versus one snowball's chances.

[qazplm135]

80. I have no idea

the intent or purpose of your response to me.

[TheBlackAdder]

36. From 1 in 302 Million to 2 in 302 Million. NJ's Pick-6 is $22.3 Million--odds of 1 in 14 Million.

[Adrahil]

9. They using "expected value" analysis.

Used in business analytics all the time.The expected value of and investment reflects the value over an infinite number of iterations. In this case, it suggests that the $2 investment is “worth it.”

[KWR65]

11. Don't bother buying a ticket. I have already purchased the winning Megamillions lotto ticket

I'm going to buy a condo in Malibu, California.

[ret5hd]

26. Prove it: what are your numbers?

Go ahead, we won't tell.

[KWR65]

34. Nice Try!

[VMA131Marine]

12. This math only really works if you buy up 302 million tickets to cover all the possible combinations

You would spend $604 million for those tickets so the prize, assuming you were the only winner, would nearly triple your money (if you took the prize in 20 annual payments. If you use a typical discount rate of 5% then the NPV (Net Present Value) of 20 payments of $80 million each is $996,825,668 so you still come out ahead even if you take the 20 year payout plan. You can deduct the cost of the tickets against your winnings, but only for the first year (I think) so this would make the 20-year payout unattractive compared to taking a lump sum of $800 million of which you would pay taxes on your net winnings of 196 million.

[liberal N proud]

44. If you purchased 302 million tickets, odds are high you will have duplicates

You most likely will not have 302 unique combinations.

[exboyfil]

57. For lower combinations in the past

lotteries, consortiums have actually attempted to purchase all the numbers. You can specify what numbers you select, but obviously in this case it is too many numbers. Given that you have to split with any other winning ticket, this purchasing all the numbers really has never made sense.

302 million is impossible though. 7 million proved to be impossible as well.

https://www.nytimes.com/1992/02/25/us/group-invests-5-million-to-hedge-bets-in-lottery.html

[muriel_volestrangler]

20. But if everyone thinks it really is "worth it" now, the chances of sharing it go up

so it's not that simple.

The UK National Lottery, when it was pick 6 numbers from 49 (about 1 in 14 million chances of winning), once had 133 people win in one draw. They won 'only' £122,510 each - less than people usually get for the 2nd prize.

https://www.theguardian.com/uk-news/2014/nov/17/national-lottery-numbers-20-years-katie-price-win-jackpot

[turbinetree]

23. My attitude.............if you don't play you can't win..........................

and if you do play and lose......................at least you tried to win...................

[calguy]

28. My attitude ..... I didn't play

and I just saved two bucks!

[liberal N proud]

45. It's like the Cleveland Browns playing football every Sunday

[Ilsa]

68. Ouch!

[NoSmoke]

24. This type of analysis is meaningless

under the circumstances and is really only applicable if a v large number of tickets are purchased e.g. if one pays $2 for a single ticket or $2,000 and the ticket wins, the rate of return on the $2 purchase is 1,000 times higher than the $2,000 ticket but the effective difference is negligible i.e in the $2 case one gains ($1.6 billion- $2) and in the $2,000 case the gain is ($1.6 billion - $2,000).

So, the bottom line is don't buy lottery tickets as an "investment" unless you can afford to buy very large numbers of tickets a few times (until you win) or a small number of tickets many millions of times.

Even so, in this particular case, if one buys 302 million tickets & $2 each (if that were physically/practically possible), a win is guaranteed so a bet of $604 million gets $1.6 billion before tax. However, the tax and the possibility that the win might have to be shared makes even this scenario pretty iffy.

[calguy]

29. And besides that......

You need to first have $604 million to even play. Now, if I HAD $602 million, why would I??

[awesomerwb1]

30. Let's make this easier

If you don't want to play, don't play.

If you want to throw a hollier than thou fit because -I- spent $2 out of MY money for MY ticket.....man, I'm just out of ****s to give.

[rickford66]

31. Why do some people only buy a ticket when the pot is huge ?

Isn't winning a million pretty good? Would they not pick up a $1 bill, only a $100 bill ? Probably the same people who wait 364 days to claim their prize if they win for some reason.

[fescuerescue]

35. If I won this, I would absolutely wait 6 to 11 months to cash in.

That would give time for the hype to die down and save at least a little bit of hoopla and negative attention.

I wouldn't give a shit about many of my current debts and stop paying them. Who needs credit when you have half a billion? Once I get the payoff, I'd catch up and just pay the penalties.

Think of all the people you say "fuck you" too, not worry about the consequences! They would have no idea.

That would be a very very easy and stress free 6 months.


But to answer your question, why don't I buy on a regular basis? Because it's throwing money away on a regular basis.. When it's this large it gets my attention and I say. Sure. Here's $2. Why not.

[rickford66]

40. You lose a year of income and you could die tomorrow.

The hoopla will happen no matter when you'd cash in. So you wouldn't pick up $1 bill.

[fescuerescue]

48. Well if I died

I wouldn't care much about the lottery.

But what I'm saying is that knowing for sure, that I was going to pick up half a billion in 6 months, would change my life immediately for the better. I would live very very well for those 6 months, without a care in the world about debt or bills...or paparazzi.

Plus it would give me plenty of time to prepare for the massive media and "friend" problem.

It's all just pie in the sky though right. But yea I pick up $1 bills, not sure what you are implying.

[rickford66]

49. "But yea I pick up $1 bills, not sure what you are implying."

You said only the big pots get your attention. So by that logic, you only pick up large bills, not the ones.

[fescuerescue]

52. Well the logic is invalid then

Since I do pick up $1 bills and I don't play the lottery except pots are massive.

Personally, I think spending $2 a year on the lottery is a luxury I can afford.

On the other hand, spending money every week on the lottery isn't something I'm interested in. It's perfectly fine if others want to, but it's not an interest for me.

[Revanchist]

37. Because a half million after taxes really isn't that much today

If the market crashes and young have the wrong investments you're screwed. We had a post a few days ago saying nursing home costs are fast approaching 100k a year so that million will get you 5 years before have to move to a shithole. If I hit the powerboat (the lower one at the moment) I'll get around 6 mil the first year and 25 mil on year 30. That's more than enough for quality around the clock care in the comfort of my own home

[rickford66]

41. You wouldn't pick up a $1 bill either.

[JI7]

42. half a million would make a huge difference in most peoples lives

not everyone is going to the nursing home.

the money could be used to start or invest in a business or something else and it could turn into more.

it could pay off debts.

[whathehell]

61. I wouldn't let the Perfect be the enemy of the Good

I'll take that half mil, taxed as it may be.

[fishwax]

70. That's a rather silly comparison, though

I only buy a ticket when the pot is huge because that's the only time it really captures my attention. I know other people like to play regularly, and that's fine, but for me I guess it's not worth the time or energy or money until there's this sort of hoopla about it.

If I see a dollar bill on the street, though, I know that I get one dollar if I spend the time and energy to pick it up (and to check quickly, depending on the circumstances, as to whether or not anyone else may have a claim to it). That's always going to be worth the minimal time and energy that it takes.

[rickford66]

75. I found a dollar today and put it into my pocket

because there was no one around. Thus has been fun.

[Hortensis]

32. :) Better hold onto your wallets around here.

I'll be glad to buy a $1 ticket if I can sell it for $5. Where do I go to do that?

[liberal N proud]

46. If someone is selling tickets of $1.00

Buy all they have. You could start a fire with them.


Tickets are $2.00 each.

[Hortensis]

63. Whoa! I've been distracted by inflation in the food supply.

Thanks for updating me.

[fescuerescue]

33. Leaving taxes out of the equation

ignores the very fact that makes it negative value.

This also ignores the fact that it's paid out over 20 years. The value of a $2 ticket today is probably closer to $5.53 over 20 years (the value of the prize WITHOUT taxes)

After adjusting for instant payout, taxes (probably close to 50% in total), a $2 ticket is still worth less than $2.

Don't even get me started on the likelihood of sharing the prize.

[joshcryer]

50. Translation: BUY A TICKET SO THE MADNESS ENDS

[EndGOPPropaganda]

51. Bad math. Prize can be split.

First even if you’re accepting the article’s premise that all that matters is expected value:
As the pot gets bigger there are more players so split prob goes up.

But bigger picture: this is not about expected value (coarsest: it’s not about math). You’re not going to repeat this play enough for the law of large numbers to affect your life.

[lunamagica]

53. Well, I have my ticket

[Cicada]

54. High Positive expected value from one in 300 million event is worthless

A positive expected value which will happen in 500 years has zero value. Payoff times probability can be extremely high but if it has basically zero chance of happening it is worthless. Because it is not going to happen.

[Quixote1818]

56. Maybe everyone on DU should pool together? If one of us wins we share with every member

who has been active this past year? I bought one. First time I have ever bought one.

[haele]

71. I got one ticket for the curiosity factor.

There's several different theories of statistical probability that one can use to justify buying a ticket, but it all comes down to the excitement factor of being part of a big event.

If you really think you're changing your odds of winning that much by buying 1 or 2 or 10 or 100 - or 1K tickets, then you're fooling yourself with the odds of a billion dollar payoff (or 1.5 billion, or whatever it's going to be).

Having been part of an office pool that only once in 15 years won anything over $120 (we won $75K one drawing split between 10 people) with an average of 20 tickets per draw, it still comes down to a probability that there is a grand prize winner in 1 in the total number of combinations per ticket. Normally, we would win enough that we could keep playing, or maybe get $5 each plus enough to buy the established base number of tickets for the pool on the next draw.

You buy 1K tickets, it's more probable that you end up with anywhere around $500 - $5K in winnings. It still ends up being a 1 in ~ 302 million chance of winning the big number, which has been proven by the fact that there has been no grand prize winner for a significant number of draws.
Which, come to think about it, is suspiciously coincidental that the pot has gone up this high without a winner. Usually around $500 million, and more people start buying multiple tickets. Who's running this lottery? How much is their cut of the winnings?

Haele

[Hav]

72. Buying 1000 tickets does increase your chance to 1/302000. Sounds good?

Last edited Wed Oct 24, 2018, 04:35 AM - Edit history (1)

Looking at the odds is the sobering reminder of how futile the situation still is. You have 1000 chances to win but there are 302million - 1000 outcomes where you don't win. This shows how deceiving multiplying your chances or even increasing your chances to 1/302000 is when you start with a ridiculously low probability to win. It may sound good, but you still have almost the same 302 million outcomes against you. The reason is, 302000 still represents all the possible outcomes. It's just that a single element of the 302000 now corresponds to 1000 different combinations each.

[haele]

74. Exactly. Probablity math is not the same as, say, linear algebra.

Each ticket has the same probability of winning. But adding them up does not change the probability, and that's where people can't see the big picture. It's the old "Let's Make a Deal" odds exercise. Given three doors, are your odds of winning greater if you pick another door once they show you a door without a prize?

So, one ticket, one thousand tickets - it's always the same odds.

Haele

[Cuthbert Allgood]

78. The "Let's Make a Deal" exercise shows it is better to switch if they open a door.

You have significantly better odds switching your door.

More tickets does give you a better chance of winning. Minuscule, of course, but better.

[marybourg]

76. I wouldn't buy a ticket, because I'd be afraid

that I might actually win. That would be the end of life as I know it,and not in a good way.