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Update See the comments, but thanks to readers’ arguments I believe I was wrong in an essential calculation.
The calculation is simple: for each ticket, multiply the probability of winning each amount by that amount, then sum these together. What is missing from the analysis is the amount the winner of a large sum would have to pay in both state and federal taxes. As a general rule, it is very dangerous to quote formulae without understanding what they mean or why you’re using them.
But one error does not fix another, so what I’ll do is to rework the text with this interpretation in mind, highlighting the strangeness of the Take Five.
There could well be an error in the reported figures, but I wonder how many people bother claiming free tickets, and whether the Lottery are happy to bet on that.

Don’t forget you get to keep that nickel you used to scratch the ticket too, always a winner! The thing I want to know is what ticket number is most common for the jackpot winners to be printed on for the scratchoffs? Enter your email address to subscribe to this blog and receive notifications of new posts by email.
Benjamin FranklinThose who would give up Essential Liberty to purchase a little Temporary Safety, deserve neither Liberty nor Safety. Incidentally, according to this source, New York State Lottery pays out an average of 61 cents for every dollar wagered. Not that it’s relevant to this game, but it hints that many NYSL games do not pay out at a rate as high as this particular scratch card.

Because if you only look at one batch of tickets, the player is partially exposed to the payout-percentage twice.
But, if you as a person buy all the 546,440 tickets in the batch, the free tickets get mixed up as you then look at more than one game-round. Imagine there is no free-play, but instead if you win $1, a man puts a gun to your head and asks you to play again. The payout-percentage is still 97.1%, but because you essentially played again you lost more and thus the lottery made more money.

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