The Powerball jackpot has hit $600 million, passing the point where people who don’t typically play the lottery decide to jump in, shell out a few dollars, and try their luck on an infinitesimally small opportunity to join the 1%
One of the most interesting ways to look at this is through the statistical concept of expected value. How much is it worth to play, given the estimated winnings and initial up front costs?
If we multiply the probability and payout of each possible prize, and sum that for all events, we get the expected value of playing the game.
When we look at a very basic expected value calculation for the Powerball lottery, we notice that playing the game is expected to be worth $1.78, given an estimated $3.78 payout on each ticket and a $2.00 investment:
However, there are a couple things to keep in mind. First, notice that 90% of the expected value is derived from the outrageously high jackpot.
Next, there are a few assumptions in this estimate that don’t mimic reality.
- The estimate assumes that there is only one jackpot winner. When there is more than one, the winnings are split.
- The estimate assumes you take $600 million in the instalment plan rather than $376.9 million up-front check all at once.
- The estimate assumes there aren’t taxes on the winnings. There absolutely are.
So let’s dissect these probabilities.
The first one — the multiple winners problem — is by far the most interesting.
When there is more than one winner, the value of the jackpot decreases because the winnings are split:
The probability there is more than one winner is directly related to the number of tickets in play. The more tickets that are in play, the more winners that are likely.
Here’s how were able to calculate the probability of the number of jackpot winners. We use the binomial distribution:
Where ‘n’ is the number of people playing in the lottery
And ‘k’ is the number of winners
While ‘p’ is the probability of winning the jackpot:
The probability ‘n’ out of ‘k’ people win =
P(n,k,p) = (n!)/[k!(n-k)!] x pk x (1-p)n-k
If that’s too complicated, don’t worry. Here’s the result for various values of the number of winners (k) at various quantities of tickets (n) sold:
Here’s a better way to look at it. You’ll notice that it’s very likely as more and more tickets are sold that there is more than one winner:
So it’s really important that we account for this when calculating expected value. When 3,000 tickets are sold every minute, the chances for multiple winners are quite good and make a major impact on expected value.
The more tickets that are sold, the higher the probability of multiple winners.
The higher the probability of multiple winners, the higher the probability of a jackpot split.
The higher the probability of a jackpot split, the lower the expected value of the jackpot for an individual ticket holder.
Since almost the entire expected value is derived from the jackpot, splitting the jackpot multiple ways is a big issue.
Here’s what the expected value of playing Powerball looks like as the number of tickets sold rises:
This decline is the direct result of a rising probability of a winnings split. Notice how at 1 player, the expected value is equal to our original calculation.
That’s because it assumes there is no possibility of multiple winners at that point.
It’s the idea that a man walks up to you, offers you these odds, and you get the payout.
As the number of players rise, the probability a winner will have to share their winnings rise.
So depending on how many people participate in the lottery, you can find out your expected winnings.
Now let’s look at the expected value of a lottery ticket given that there’s the possibility for multiple winners and you decide to take all the money up front — a $378.9 million check instead of $600 million over time. It’s potentially split between winners.
Here’s how that looks:
Once more than 200,000,000 tickets get sold, it’s game over. Assuming a split check taken immediately, you’re cooked once more than 200 million tickets are sold.
Finally, we’ve got to incorporate the tax man’s impact on your winnings. As a warning, it’s not going to be pretty.
Assuming an instantaneously awarded check for winnings, you’re going to be in the top bracket no matter what. Even assuming that the person makes no money, applying the marginal tax rate to the check — all while assuming multiple winners are possible — makes this picture look even worse:
So what’s the moral of the story?
Buying a Powerball ticket as an investment makes sense, as long as you only look at the simple case.
When you analyse it deeply, you’ll see that even after you account for multiple winners and potential split jackpots, it’s still worth it, just not as drastically.
But once you consider taking all the money up front rather than the annuity, you’re going to see a problem. After 200 million bought tickets, you get a negative expected value, and it stops being worth it.
Then, after you consider the federal government fleecing you with a 39.6% top tax rate — welcome to the 1%, we hope you enjoyed your stay — it’s pretty obviously not worth it.
And we didn’t even look at state income taxes.
So is it worth it to pay the lottery? That depends. If it’s worth it to spend $2 on several days of wishful thinking, there’s nothing stopping you from playing. While it’s rarely profitable overall, it’s fun to gamble, that’s why people do it.
But don’t look to maths for a thumbs up.