In the All you mathematicians, get your butts in here! thread, SingleDad said:

Now, SingleDad could be talking about two classes of misconception. The first, really a mathematical triviality, is the simple neglect to consider the effect of taxes and the time-value of an annuity pay-out (i.e. “a lottery ticket has a positive expectation where the *net present value of the after-tax payout* is more than X where the odds are 1/X”). The second, which raises interesting mathematical/statistical issues is the possibility that the payout will be shared among several winners.

My question is, given rational assumptions about net after-tax cash-value payout, chance of winning, number of participants, likelyhood of multiple winners and all other relevant factors, does the purchase of a single ticket (or low number of tickets relative to the number of combinations) have a positive net value.

I know that arbatraging the lottery (buying all numbers) will be a losing strategy (particularly if there are multiple arbatageurs).

However, let’s say that the odds of the NY Lottery are one in 13 million, and that the cash value payout is about half of the advertised payout, and that you’ll pay taxes of about half of the cash payout, so that (assuming you will be the only winner) a payout of $50 million would have a “positive” value.

Now, even if there is a possibility that there *could* be multiple winners, it seem that there are never (or exceedingly rarely) more than two winning tickets on the big lotters, and usually there is only one, so under most circumstances there would be a “positive” value to buying one ticket in a $100 million lottery.

(For the actual rules of Lotto in New York, go to http://www.nylottery.org/ .)

How would we determine what the amount of lottery payout is big enough that it’s a good bet to buy a ticket?