In case you guys haven't seen, the Mega Millions currently has a jackpot of $476 Million, which is really a cash option of $341.1 Million.

The odds and payouts are here

Now, if I'm not mistaken the way we calculate this is take the payouts over the odds as fractions and add them up, yes?

E= 2/74 + 3/140 + 10/843 + 7/346 + 150/13781 + 150/15312 + 10,000/689,064 + 250,000/3,904,700 + 341,400,000/175,711,536

Look good?

If I do my math right, before the jackpot (ie the part that never changes) a ticket appears to have an expectation of about 18 cents (E=.0.1798).

Currently, a ticket is worth about $2.12 (E=2.1228) making the ticket a screaming buy! Looking at this another way, if you would buy all 175M+ combinations you'd more than double your money!

Also, the breakeven point on the Mega Millions is when: (1-0.1798) = (X/175,711,536) the cash jackpot option equals $144,118,601

Did I make any errors?

*this all ignores the effect of splits and taxes

I think it will be +EV, before taxes and after chance of split, in the range of $1.15/$1. However, buying all 175m tickets would not be a +EV strategy as that would put the number of tickets purchased over 365m... I wrote up some more at the link below on a site I never really got around to doing anything with, but this big jackpot prompted me to do a little write up.

http://lotteryev.com/mega-millions-lottery-math/

With taxes, and jackpot split chances, it probably will not be +EV still.

Even rwesty's post above ~proves that. $1.11/$1 before taxes = -EV after taxes.

But is it worth throwing $10 or $20 on to change your life? **** yeah.

If it rolls over AGAIN, it might be +EV even post tax.

Well, the effect of a split is pretty easy to figure out. Just multiply the jackpot size by 0.5.

Your link won't work for me. I'm not getting what you're saying about the number of tickets purchased being >365M.

assume n is the number of ticket number possibilities
assume k is the number of tickets sold in a drawing

if you multiply the jackpot by (1-e^(-k/n))/(k/n) that accounts for the possibility of having to split the jackpot with another winning ticket. this is a big factor when there will probably be around 300m tickets sold for friday's drawing.

if there are more than ~365m tickets sold it will be -EV even before taxes.

Yeah, but I have 1 of every combination, so I will win once and lose on all the other times. So the 175,736,511 tickets I buy shouldn't really be factored in.

Furthermore, my buying those tickets raises the jackpot by 32.5% of the money I spent, 57,114,366. Even sweeter.

I really wish that you could ignore splits and have a guaranteed jackpot so I could walk in to 7/11, drop my $175,736,511 on the table and say "That'll be $374,283,621 please!"

yeah, i didn't think about how buying more tickets does increase the jackpot as well. not sure if the lottery gives out 100% of the information necessary to calculate that or not and you certainly wouldn't know exactly how you were going to be affected until after the drawing... also, does that depend on how many people happen to win the 5/5 w/o the 6th ball?

Splits are hugely important. I adjusted for taxes and splits and did the math according to http://www.durangobill.com/MegaMillionsOdds.html and I got that this jackpot will be slightly +EV as long as more than ~280 million tickets aren't purchased. FWIW, there were approximately 180-190 million purchased for the Tuesday drawing. With the same amount a $1 ticket bought for this drawing would be worth about $1.17.

~ 31.8 cents per dollar goes to the jackpot, 18.2 cents to all other prizes (you have rounding error Jay on the smallest payouts), and 50 cents to the government. Buying up every combo should up the cash option jackpot by about $56M.

With the new jackpot estimate now at $500M annuity ($359M cash), I would estimate that they are estimating that $330M in tickets will be sold for this draw. I also have a feeling the sales number will creep up a bit more by Friday though.

If $330M in tickets are sold:

Rollover (holy **** if this happens): 15.29%
1 winner: 28.71%
2 winners: 26.96%
3 winners: 16.88%
4 winners: 7.93%
5 winners: 2.98%
6 winners: 0.98%
7 winners: 0.25%
8+ winners: 0.074%

If the jackpot would be hit, the effective payout of the jackpot after prize splitting probabilities will be 59.7%; $214.4M cash. This leads to $1.22 jackpot value per combo. Lumping in the other prizes (~18.2 cents): $1.402 total ticket value. This is all pretax of course. Taxes still wipe away the profitability of playing.

Taxes (federal + state) will wipe away ~40% of all but 10.35 cents of that $1.402 per ticket value (10.35 cents per dollar goes to prizes $150 or less), so with taxes, then we can estimate the return to be about 0.6*$1.2988 + $0.1035 = $0.883.

Still -EV after taxes. But if it rolls over again...

My friend argued with me earlier about this using some weird sort of opportunity cost approach and I am pretty sure there is no chance the Mega Millions right now is +EV and therefore not playing is a better option than playing based....Is this proper thought process?

- $341,100,000 cash option
- 300,000,000 tickets sold (I know this isnt absolute number but for calculation sake I just went with this)
- 175,711,536 possible combination of numbers
- Using binomial distribution to determine multi-winning tickets got ~53%
- After-taxes you are probably only keeping 64% of winnings due to highest tax bracket?

So (341,100,000/175,711,536) x .53 x. 64 = ~$0.66 for jackpot only and then adding the other bonuses that you can hit without the "mega" number is going to put you only at a little less than $0.80 per ticket I think?

Does the fact that many of us file as professional gamblers have an impact on these after-tax calculations?

yes, if you have other gambling income that already puts you in the highest tax bracket you can entirely ignore taxes in your EV calculation

wait wat?

This is such a pimp business for the states ... they pay out 45 cents on the dollar and then they tax 45% of the 55% . just a cash cow for them... as it stands,, the chance of a split happening makes this a -ev play still.

Ya, obviously our time and capital would probably we better spent playing poker. But, for the casual gambler who only plays the lotto and scratch tickets this is prob their best spot in a long time.

Most people give up another ~5% for state taxes too I think.

lol.....jesus.

because if you lose that dollar you save the same percent on taxes as you would have to pay if you win

What about being able to net your winnings. I assume this has to have some type of positive effect ... because if we won we could play in some high roller events or the million dollar one-drop and then simply deduct the buy-in from the net. Also, say we bought 150,000 tickets .. we could simply deduct that from the winnings. But if a casual player wins (and doesn't change his filing) then he has to pay taxes no matter how much he spent on the tickets and no matter what else he gambled on ... correct?

no, if they bought x amount of tickets and they itemize deductions , they can net it out .. say they won 350 million but bought 180 million tickets..
they pay taxes on the 170 million.

gambling tax laws can be weird, for non-pro's especially, and are different in every state... but in most states i think non-pros could also deduct tickets bought... but yes, i think there would be a benefit to filing as a pro already

Slight thread hijack: is my intuition right and NOT choosing the multiplier options is the highest ev for the friday drawing?

Definitely do not purchase the Megaplier. I think the return for that bet is only about 48%.

The saddest thing about that return is it is often better than the overall lottery (when the jackpot is under 60m), so purchasing it is actually a better option than buying twice as many tickets.

If this site is correct, it looks like at most 40% of ticket sales go into the jackpot. Since they put the jackpot at $500MM, that's an increase of $137MM over last jackpot which means they're projecting 343MM tickets sold, and that 40% may be high. So, way too many split pots to make this +EV.

Based on those numbers, the past 4 drawings have contributed .38,.44,.53,.44 to the jackpot. If we assume the average (.45) is contributed there will be 310m in sales. If we assume .38 there will be 360m. Either way the drawing is still +EV between .05-.15c/$1

Assuming the average of .45 is contributed, this drawing will remain +EV until ~470m tickets have been purchased.

I expanded a little more here - http://lotteryev.com/more-about-frid...lions-jackpot/