Editor’s Note: This post was previously published – With the Powerball jackpot reaching $550million for Saturday’s drawing, what better time than now to refresh your memory on what the lottery odds actually look like…
This week’s Powerball jackpot is set to be a record $425,000,000. This prize is paid as an annuity over 29 years. The cash value is estimated to be $278,300,000 if the lucky winner decides to take a lump sum payment.
We all know that the lottery is a sucker’s game, “A tax on people who suck at math.” as I’ve often heard it put. But it seems every few months the Powerball or Mega Millions lottery jackpots reach these astronomical amounts, driving even the most hardened skeptics (read: me.) to run out and buy a ticket.
When the lotto jackpots reach these gigantic numbers the question I always wonder is, “How big does the jackpot need to be for it to be a good decision to buy a ticket?”
To put it another way – does a lottery ticket ever have a positive expected value (+EV)?
Expected Value is simply the average amount you can expect to gain or lose on an event given the average results of each possible outcome.
To calculate Expected Value (EV) we use the following formula: E[X]= x1p1 + x2p2 + x3p3 +… + xnpn
For those who aren’t mathematically inclined, X is a random variable that can have values x1, x2… with the corresponding probabilities of p1, p2 and so forth.
Say you go to a casino and want to bet $1 on a single number at the roulette wheel. The roulette wheel has 38 spaces and the casino pays out 36:1 when your number hits. So if the ball lands on your number you win $35, if it lands in any of the 37 other numbers, you lose your $1 bet. The EV of that $1 bet would be:
E[$1 bet] = $-1 * 37/38 + $35 * 1/38 = $-0.053
So you are losing just over 5 cents every time the dealer spins the roulette wheel with your $1 on your “lucky” number. Vegas, baby!
Now back to the Powerball jackpot…
A Powerball ticket costs $2 and the Powerball website lists the odds of winning the jackpot at 1 in 175,223,510. However, there are also smaller prizes available for hitting fewer numbers. The odds and expected return for the different prizes looks like this:
The sum of the probability column is 3.14%, meaning that approximately 1 out of every 31 tickets will win a prize of any kind. The total of the return column, 1.95, means that for every $1 risked in the $278 million jackpot you can expect to get back $1.95. Earning 95 cents on every dollar wagered in the Powerball drawing is a really good outcome. However, as you might be able to guess. These numbers are too good to be true.
For starters, we’re completely ignoring the fact that there can be multiple people with the same numbers, thus causing the winners to split the jackpot if their numbers hit. As the jackpot grows, more and more people buy tickets which increases the likelihood of a split pot.
Secondly, and most importantly, this example completely ignores taxes! Applying the top federal income tax rate of 35% to just the top two prizes decreases the return to 79 cents on the dollar. Even with the largest Powerball jackpot in history, federal taxes alone cause the lotto player to have a negative expected return on his/her bet.
Even with record jackpots it’s safe to assume that the odds of turning a profit on your Powerball or Mega Millions ticket won’t be in your favor.
From a purely mathematical standpoint, the lottery isn’t worth playing. But on the rare occasions I play the lottery, I don’t play because it’s a +EV move for my money. I play in case the Sun, the Moon and the stars align and my numbers hit that $278 million prize.
I don’t play the lottery with money I can’t afford to lose, and I don’t expect to see a positive (or any) return on that money. I just dream about what I’d do with all that money, and hope that maybe it’s my lucky day.