Wednesday, January 13, 2016

Mathematically Rational To Buy A $1.4 Billion Lottery Ticket

America has lottery fever, and the only prescription is more Powerball tickets. The jackpot for the nation’s largest lottery grew to $1.4 billion over the weekend after Saturday night’s drawing produced no winners. So with the jackpot now at ten figures, has it finally become mathematically rational to purchase a Powerball lottery ticket?
Not quite, but it’s getting pretty close. In order to understand why it still doesn’t make pure mathematical or financial sense to purchase a $2 Powerball ticket, you really only need to know about two factors affecting the winner’s ultimate payout: state and federal income taxes, and the possibility of a split pot.
Before we get to those two issues, though, let’s briefly recap how Powerball works and how the basic probabilities of winning are calculated. To win the jackpot, you must get the correct red Powerball number and match each of the five white numbers. The white numbers don’t need to be in any particular order, just so long as you have all of them and the Powerball number (in geekspeak, the Powerball probabilities are calculated using combinations, not permutations). The winning lotto combination is drawn from 26 red Powerballs and 69 white balls. There are a total of 292,201,338 different Powerball combinations, but only one number can win the jackpot.
The jackpot, however, isn’t the only way to win money. You can win anywhere from $4 for guessing just the Powerball number correctly to $1 million for correctly guessing all of the five white balls while missing the red Powerball number. The probabilities of those different incomes obviously affect the game’s odds and slightly change the expected payout if you choose to purchase a $2 ticket. You can get a full run-down of the probabilities and the underlying math here if you’re interested. While the probability of winning the jackpot is exceedingly low—0.0000003422 percent, to be exact—your overall probability of winning something is just over 4 percent.
Powerball Probabilities
So with that math out of the way, we can get down to business calculating the expected payout of a ticket. In mathematical terms, the expected payout is a function of the total payouts for each scenario or outcome and the probabilities of each of those outcomes.
For example, imagine a game where you flip a fair coin, which would be a 50/50 proposition. If you get heads, you win $1. If you get tails, you receive $0. The expected payout of that game is $0.50: 50% x $1 for heads ($0.50) plus 50% x $0 for tails ($0.00), for a total of $0.50. As a pure mathematical matter, a risk-neutral person would happily play that game if it cost less than $0.50 to play.
Why? The net expected payout—the expected payout minus the cost to play—would be positive. If the cost to play exceeds the expected payout, it does not make financial sense to play. Because lotteries almost always have a ticket price that exceeds the expected payout, it rarely makes financial sense to buy a ticket.
In the case of the Powerball lottery, the house consists of the federal government and various state governments administering the lottery.
The lottery, much like games at any casino, is heavily tilted in favor of the house and against the player. In the case of the Powerball lottery, the house consists of the federal government and various state governments administering the lottery. That’s because of taxes, the possibility of a split pot, and the fact that the advertised jackpot will always be significantly lower than the ticket revenue collected.
While the current lump sum cash payout being advertised by Powerball is $868 million (the eye-popping number usually advertised is the annuity value paid out over 30 years, not the immediate lump sum payout), that’s not how much will end up in your pocket if you win. You’ll send 39.6 percent of that right back to the federal government in income taxes. And depending on which state you live in, you’ll have to send another hefty check to your state government.
As a result, instead of using the advertised lottery payout in our expected payout calculation, we have to use the after-tax payout to determine whether it’s really a good financial idea to buy a Powerball ticket. To make things simple tax-wise, we’ll assume you live in a state with no income tax and that you’ll pay the top federal marginal income tax rate of 39.6 percent. Using those assumptions and current jackpot cash payout estimates, you’ll take home $524 million after taxes.
We can’t stop there, though. Because more than one person can win the Powerball jackpot, we have to take into account the probability of a split pot. The chances of a split pot increase as more and more people purchase lottery tickets. According to Powerball officials, over a billion dollars worth of tickets were sold over the weekend, meaning at least 500 million tickets were sold to hopeful buyers. If you use the split pot probability table shown below and use 500 million tickets as your benchmark, the probability of a split pot reduces the expected take-home amount by roughly 37 percent. So that $524 million after-tax expected payout needs to be bumped down to $328.6 million before we can calculate our expected payout.
Split Pot Payouts
To get the expected after-tax jackpot payout, you would then multiply that number by the probability of winning the jackpot, which is 1 divided by 292,201,338. What’s the end result? A whopping expected payout of $1.12…on a ticket that cost you $2. Once you add in the expected payouts from all the other potential Powerball prizes, that number increases to $1.32. And again, that assumes you live in a state with no income taxes.
So right now, given the advertised cash payout and the number of people playing, it does not make financial sense to purchase a Powerball ticket. That said, we’re not all that far off from a jackpot that might justify the purchase of a ticket. If the lump sum cash payout (not the annuity value) increases to $1.4 billion, and the number of tickets sold hovers around 500 million, the total expected payout of a $2 Powerball ticket will be $2.01, making the purchase of a ticket a rational decision in pure mathematical terms.
If we go another week or two without a Powerball winner, resulting in higher and higher lump sum jackpot amounts, we might finally reach a point where it actually makes financial sense to buy a Powerball ticket.

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