Mega Millions Expected Value Calculator

Presented on this page is a calculator which allows you to enter some details about an upcoming Mega Millions drawing and receive the tool’s best guess of the expected value per ticket purchased. For details, read the article below the calculator.

Mega Millions Ticket Sales and EV Estimator
Input Data Values
Enter Jackpot
Net Present Value (decimal, to convert prize to cash value)
Federal Tax if You Win (Percentage)
State and Local Tax if You Win (Percentage)
Expected Value of Sub-Prizes, Pre-Tax, Dollars

Mega Millions Predictions
Calculated Results Values
Predicted Tickets Sold
Expected Value of Ticket (Dollars)
You Are Wasting This Much Money Per Ticket (Dollars)

Code created with assistance from Political Calculations

Methodology

If you don’t care, scroll down to the conclusion!

Building on the work advanced by Jeremy Elson in this article on Mega Millions Odds, I decided to take his work a couple of steps further. I fleshed out this calculator which could both estimate the number of ticket sales for a given prize amount and, using a guess of ticket sales, give a prospective ticket buyer an estimate of the expected value of a single entry. Roughly, here’s the steps I took to create the calculator above:

1) I went to LottoReport.com and downloaded all of their information on Mega Millions drawings since California joined in 2005 – a total of 789 drawings, as of this article.
2) I calculated the expected value of the sub-prizes (see my previous article, in which I originally had the ticket sales wrong – but the methodology right).
3) I calculated the expected value of the prize based on the Poisson distribution n=1 through 12 (again, see my previous article on lottery odds) to approximate the distribution of winners, and used those odds to guess at the winner’s share.
4) I took two data sets to ZunZun, the online curve fitter: one which used the 789 prizes and ticket sales, then another which used the 789 prizes with the calculated expected prizes (I also weighted this data – I used ‘Prize amount divided by 10 million’ for weights since there was a dearth of data on the upper end of the scale). For curiosity, ticket sales worked best with a Peters and Baskin Step Stool function, while expected prizes followed a rather tame Double Rectangular Hyperbola.
5) I put them in calculator form. For expected prizes, I took it a step further by dividing by odds (1 in 175,711,536), then finding NPV and the post tax EV.

Weaknesses

Most of the weaknesses in this strategy come in using the Poisson distribution to approximate tickets sold. Yes, Poisson is great in a vacuum, where everyone uses random picks. However, being humans with influence of control, many people like to pick their own numbers to enter in a Mega Millions drawing. This means that entries aren’t truly random and might follow some exploitable pattern based on favorite numbers, birthdays, or something else. For further discussion on this point, see my colleague Cameron’s article on calculating lottery odds. Oh, and if you figure it out, you might find that the Mega Millions is exploitable. Come back here and share your results (after you test it, of course!).

A second weakness is in the ticket sales. I expect ticket sales for the smaller drawings to be pretty accurate – those sizes happen quite often. However, tickets sold in some of the record setting drawings start to become unhinged from the neat progression of the smaller ones – as hype builds and the lottery gets media mentions, sales become self-fulfilling. We have no idea what happens if we approach a \$640 million drawing again – perhaps buying is more tame next time since we’ve seen it before? For one example, note that the \$363 million drawing on 3/27/2012 had 190.9 million tickets sold for that drawing, but the \$380 million drawing on 1/4/2011 drew 229.4 million. Could the next \$640 million jackpot be ‘old news’ and therefore draw less ticket sales? (Yes, the lottery itself will increase prizes in reaction to sales, but it is still possible for ticket sales to come in less than this model). This is a ‘less weak’ point than the above, but still something to think about.

Conclusion

What did you expect to find? Did you think the lottery was a good deal?

Not only do we find from reasonable assumptions that the Mega Millions hasn’t been a good deal, but at a certain advertised prize size around \$450 million, the ‘climbing EV’ starts to reverse and larger prizes are actually a worse deal. Fancy that (see nearby graph).

Except in very specific situations (You live in a high tax state and have enough gambling winnings to write off losing tickets), you never had a positive EV drawing for the Mega Millions – at least since California joined. Go run the data yourself if you think it’s there before 2005 (doubtful…). Still, I think Greg at Control Your Cash said it best: “The 3 people who won can make fun of us. The other dozens of millions of losers who bought tickets can grab a piece of this and slide off.”

Indeed, except for gamblers from New York who were able to offset their winnings. You’re safe… this time.

1. says

You mention that humans used their favored numbers as exploitable.  Exploitable as in from the lotto organizer side?

• says

No, exploitable from a ticket purchaser side. I’ll explain it better:

Say you had ‘almost’ positive EV. One example would be you live in Texas (no state tax), NPV is .77 (it is actually around that this year – .63 is an older number), and the jackpot is \$480,000,000. On the surface, with a perfectly random distribution – your EV would be .88 per ticket… so, still a bad deal.

However, one thing we ignored in the calculation is that about 30% of Mega Millions tickets are manual picks. From just the ‘random’ tickets (70% of the total – around 247 million) our expected prize is about \$328,241,740… so if you removed the manual picks, our EV would be that number over 175,711,536 (jackpot odds) plus .18189 (lower prizes) x .77 (NPV) x tax = \$1.01 per \$1 ticket. So, if you ‘knew’, for example, that people would never pick numbers above 31 or avoided 13s or didn’t like odd numbers and that edge stuck around? You could buy tickets using that strategy and actually have it worth your while.

Still, I think it’s a long shot for the Mega Millions. That doesn’t mean that people haven’t hacked lotteries in the past – I just think the exposure Mega Millions gets means it is unlikely you will ever see an edge.

So, incredible luck aside… I would avoid it, haha. (Do as I say, not as I do?)

2. Nice job making up a calculator. I know that whenever I buy a ticket I’m just wasting money because it’s almost a guarantee that I won’t be winning the jackpot. Still, it’s fun to pretend for a couple days that something wonderful might happen, and it only costs a couple bucks.

• says

Fair enough, but I did want to popularize the fact that it’s not in your favor, heh. But hey, it was interesting to buy those \$60 tickets the other day – funny story losing on every one, at least!

3. says

You might find this story interesting:

“As an example, one RNG with which I am familiar is that used by Dr. John
Gwynn in testing blackjack systems. His RNG has a cycle length of 2.7
billion. This means that his RNG will produce a string of 2.7 billion
numbers which exhibit no discernible or predictable pattern, random for
all intents and purposes, but that the cycle will begin repeating itself
after 2.7 billion numbers.

“Without the clock chip to generate different seeds, each day the
Montreal Casino was cycling through the same numbers, beginning at the
same starting point! This is what Daniel Corriveau discovered. And this
discovery paid him \$600,000 in keno winnings.

“Incidentally, Daniel Corriveau was paid his \$600,000 after investigators
determined that he did not work in collusion with any casino employees.
He took advantage of the game exactly as he found it. Hats off to Mr.
Corriveau for teaching the Montreal Casino a lesson in mathematics. If
there is a way to beat a game, someone will find it.

♠

• says

You’re really going to love this Kelly Criterion and Ed Thorp stuff I have in the works! Nice link. I have this one for you, but from a different perspective (RNG for cryptology):

Dual_EC_DRBG

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