Update See the comments, but thanks to readers’ arguments I believe I was wrong in an essential calculation. I’ve reworked the text to correct my error, which was small in most cases—except in one important place. But in making this error, it reveals that the New York Lottery has one gamble which is near enough a sure thing (at least to break even), a very odd situation.
Update 2 See HaroldW’s comments: the tickets aren’t the same!
I received a call from Noelia de la Cruz at Business Insider for an article she was writing about scratch-off tickets at the New York Lottery, “We’ve Figured Out What New York Scratch Ticket Has The Best Payout.”
The easiest calculation is the amount you will probably lose. This is the ticket price. So if you buy a $1 ticket you’ll probably lose $1. If you buy a $30 ticket (yes, they have one), you’ll probably lose $30. And so on for the other ticket face values.
In other words, you’ll probably lose. Unfortunately, this statement is logically equivalent to “you might win.” And somehow this is all some ticket buyers ever hear. The mere possibility of winning, no matter how improbable, is all the encouragement that these people need.
It should come as no surprise that the odds are against you. But if you are going to gamble, wouldn’t you like to gamble on the game which has a best payout? Here’s how.
The New York Lottery has nearly four dozen different “games”, i.e. gambles, you can play. These are ever in flux. While we were preparing this article, two gambles went out of circulation and two more came on line. The NY Lottery website lists the current gambles. Each ticket has an explanation of the odds of winning various payouts plus the odds of winning at least one of these payouts.
For example, the $1 Take Five ticket allows you to win $1 which has odds 1 in 8.82, $2 at odds 1 in 23.17, and so on up to winning $5,555 at odds 1 in 529,200. You can even win a free ticket which has odds 1 in 2:84. So when you buy a ticket, you might win nothing, or you might win $1, or $2, and so on. You therefore have a chance of winning nothing or at least one prize.
That you can win a free ticket means something special. Stay tuned for what.
The formula you need to convert odds to probability of winning is this:
probability = 1/odds.
The probability of winning $1 on this ticket is thus
probability = 1/8.82 = 0.113,
which is just over 10%. The probability of winning the “jackpot” is 0.0000019, which is about 2 in a million.
The webpage lists the probability you win something, for this ticket: “Overall Chances of Winning: 1 in 4.64”. Thus, the probability is 0.216.
But the lottery erred in printing this number. This is the probability of winning money for just this ticket. But since you can win a free ticket (with probability 0.26), you might win money on this free ticket, too. The chance of that is 0.216, as well. Of course, this free ticket might win you another free ticket, which gives you another chance to win money—or another free ticket! And so on to infinity.
The overall chance of winning something is then 0.291 and not 0.216 (this is odds of 1 in 3.43). Which means you have a 71% of losing.
The ticket with the highest chance of winning is $30 Win $1 Million A Year for Life: 0.341. The $1 Take Five has the second highest. The lowest, at 0.195, is the $5 Cashwords ticket.
There is another calculation relevant to gamblers and the lottery: the payout. Now, scratch-off tickets are a kind of modified raffle. The Lottery knows exactly how much money they are going to give away (on a rolling basis, anyway) and just how much they will take in (if all tickets are sold). These two numbers, modified by circulation statistics, are all that is needed to calculate the expected payout to set the odds of winning each prize.
Since the pool of money is fixed by the Lottery, the expected payout is known, i.e. not random, to them. But it still is to the gambler, because of course he doesn’t know how much has already been won and how much is left in the pool. Therefore, at least as a rough guide, we can calculate the “expected payout” for each ticket. Think of this as the rate of return.
The calculation is simple: for each ticket, multiply the probability of winning each amount by that amount, then sum these together. For the Take Five, this is $1 x 0.113 + $2 x 0.043 + … + $5,555 x 0.0000019 = $0.65.
Here is where the mistake reveals something strange. We still have to account for the free tickets, so the final expected payout is $0.9995, that is, $1. I.e., you will break even on this game (on average). The New York Lottery is thus losing money on this ticket, after considering their light bills, shares to retailers, and so forth.
It was this strange finding that convinced me my original calculations were correct. I could not believe the lottery would create a gamble where they effectively lose money. But two readers have convinced me that this is so.
The second highest expected payout is $0.93 (per dollar) for the $30 Win $1 Million A Year for Life ticket (I assumed that in all “win for life” games you lived 40 years more). Most payouts are in the 60 – 70 cent range (per dollar gambled). Meaning the Lottery takes about 35 cents gross profit per ticket.
The lowest payout is the $2 Money Match gamble at $0.62 (per dollar, or $1.24 for the ticket). Interestingly, this ticket has the most—37!—different ways to win ($1, $2, $3, and so on). I’ll leave it as a homework problem to show that as the number of ways to win increases, the expected payout will decrease.