To me, it seems to be counter-productive for people to play the lottery only when it’s reached an astronomical amount. I think you’d have better odds playing when the amount is smaller, since there’d be less competition. I mean, I’d be perfectly happy being the only winner of a couple million dollars, as opposed to sharing a larger amount. But at a certain point, the total amount is so large that even sharing it with X number of other people, is still an astronomical amount. So historically, based on data of past results, is there an optimal amount at which you should play, and sit it out the rest of the time?
Actually, statistically speaking, you come out ahead only when the jackpot becomes humongous. The expected value of a ticket is now greater than its cost. Intuitively stated, you could be guaranteed a win by buy each of the (IIRC) 170 million possible tickets. So whenever the take-home jackpot is bigger than $170 million, you come out ahead. (Of course, you have to figure out the odds of splitting the jackpot, which would dramatically reduce the expected winnings). Sometimes there are “investment groups” that buy large numbers of tickets whenever this happens.
Still, it’s a very high-risk, since the actual odds of winning anything are still tiny. But if someone made a habit of entering high-jackpot lotteries when the expectation is positive, and purchased a large fraction of tickets each time, they’d come out ahead in the end.
The odds of winning a lottery don’t change in relation to the number of participants - it’s not a raffle. Even if you’re the only person to buy a ticket, the odds (in the UK anyhow) are still approx. 14 million to 1 against you winning. So it only makes sense (well, it never makes actual sense, but it’s better) to play when the potential prize is large enough to make the cost of playing stack up against the odds you’re facing.
Assuming all tickets are chosen at random, and the probability that a given ticket is a winner is p (very small) and N (very large) be the number of tickets (in addition to yours) sold.
Then the probability that if you win a prize you will share it with k (k>=0) winners is given by the following formula for the Poisson distribution
P(k winners)=(exp(-Np)*((Np)^k))/k!
so if you buy one ticket, that the top prize has value V, your expected winnings are
Expected winnings = Sum( (pV/(1+k)) * (exp(-Np)((Np)^k))/k!) where the sum is over all values of k from 0 to N.
In practice the later terms of the sum go very quickly to 0 so you only need to calculate the first few values of k.
This formula becomes even more interesting when you consider that V is determined by volume of ticket sales during prior drawing periods, so the formula could be modified to express values as V(N).
My extensive (okay 2 minute) poking around does not reveal the actual formula used, it would be interesting to see. I suspect that this modification reveals that at all levels playing the lotto is only worthwhile if you have a non-linear valuation of money.
The expected value of a ticket, using the Poisson distribution described by Buck, asymptotically approaches the ticket sales percentage devoted to jackpot from below.
In megamillions, the states get a pretty hefty cut of the pie, and the most frequently cited figure is that approximately 50% of ticket sale value goes to the jackpot. For large jackpots (such as the current) the expected return of your ticket is $.50. For smaller jackpots, returns are even worse.
There is however, an additional complication. Number choices are not uniformly distributed. The Poisson distribution describes a world where everyone randomly picks their numbers. You may get a slight edge if you can determine what numbers are ‘overpicked’ and base your selections to the contrary. If e.g. you can guarantee that you only buy sequences that are 50% ‘underpicked’ your expectation comes to break even. However, it seems extremely unlikely that the number distribution is that badly skewed.
So, how do you profit from the record jackpots?
Form an ‘investment group’ such as those described above and pay yourself a decent salary from the funds contributed by the non-mathematically inclined.
Have an inverted marginal utility where your value for $1 at your current net worth is half of the value you would place on $1 when fabulously wealthy.
Live in a state receiving $0.35 for every lottery ticket sold.
What percentage of the lottery revenue ® from a round is returned in prizes? I think the formula, assuming the grand prize is won, is something like
(0.6 R + S) / R
where S is the residual revenue from previous rounds where the grand prize was not paid out. Although the basic payout may be only 60%, it is easy to see that it can get above 100% when there is a substantial residue.
Of course, the effective payoff (which determines whether betting has positive value to you) has to be reduced by taxes, staggered payments and utility non-linearity. Still, if you’re going to play at all, it will be best to play when there’s a large residue to start with.
The “risk”(*) that you’ll win a too-large amount (e.g. $640 million) is reduced by the fact that with so much betting, you’ll probably just get a split-win and have to settle for a mere $100 million or so.
(* - It may seem strange to worry that a jackpot is too big, but you might feel that $64 million would have almost as much value to you as $640 million. In that case you’d much rather have ten times the chance of winning one-tenth the amount. But, as I suggest, you might sort of get that anyway, given the likelihood of a split pot.)
Aha, good catch. I believe for megamillions, the percentage is quite close to 0.5.
This also brings an interesting behavior consequence - when there is a significant residual S, more tickets are sold and R increases. While looking for ticket sales results it appears that most Mega Millions outlets are experiencing record sales due to the current hype.
I would bet that population behavior tends to strongly force S to a small fraction of R. I wonder if there’s data available to look at trends between sale rate and initial jackpot size (the S term).
Yes. Ignoring the inconvenience of purchasing tickets and taxes, etc., an efficient market would force R to approach (2 S), given vig=0.5.
In another thread some Dopers suggest the inconvenience of buying all 170 million tickets is deliberate to prevent a guaranteed win. I’d guess the opposite: that lotteries would be delighted to maximize revenue.
I know nothing of these lotteries; is there a way to buy thousands of numbers with a single ticket? (I once exchanged e-mail with a horse-track bettor, who spoke of machines that could sell 1000’s of parlay combinations with a single ticket.)
It does appear that the residual tends to be about 5-7 larger than ticket revenue during a lottery period. There appears to be an exponential relationship between payout size and ticket sales, but in observed lotteries it hasn’t been enough to swamp out the residual. It’ll be fun poking around to see what that does to expected returns.
So, I evaluated all lotteries in the history I linked earlier. I adjusted the payout by: 1. converting the reported jackpot to a lumpsum payout (0.72 adjustment factor) and 2. including a 35% federal income tax.
Using these adjustments and estimating the number of tickets sold as twice the increase in jackpot size during the lottery round, it appears that:
E($1 ticket) is strongly and positively correlated with initial jackpot size
Prior to this lottery, E($1) has never exceeded 0.7; it’s currently a bit under 0.75.
The historical mean E($1) for all lotteries is $0.17. The historical mean expected return for all tickets sold is $0.27.
That’s using the Poisson distribution. Again, if there’s a significant picking bias for the majority of tickets sold, you may be able to improve the expected return.
This is an interesting point - would be great to see some data on how much of an edge can be gained by exploiting this “picking bias”, although I suspect good data would be very hard to come by. Seems like some of the more obvious strategies would be to avoid playing numbers 1-31 (since people often play birth dates, anniversary dates and the like), avoiding sequential number combinations, etc. But of course, no idea if and by how much this could push E over the 1.0 mark.
The author makes some different assumptions (e.g. 0.63 lumpsum vs. 0.72) and higher tax rates. He also includes a figure for non-jackpot winnings, with non-jackpot estimated value at $0.1.
Including this in my more generous analysis still does not push the purchased ticket into positive returns. However, there may be another factor involved - apparently there’s some sort of multiplier that can be purchased for non-jackpot prizes. If a multiplier purchase contributes to jackpot size, I am overestimating tickets sold, and the expected return will increase for those not purchasing a multiplier.
