The Powerball lottery drawing for Saturday evening has hit a record-high estimated jackpot prize of $1.5 billion at the time of this writing.
While that’s a huge amount of money, buying a ticket is still probably a losing proposition.
Consider the expected value
When trying to evaluate the outcome of a risky, probabilistic event like the lottery, one of the first things to look at is expected value. The expected value of a randomly decided process is found by taking all of the possible outcomes of the process, multiplying each outcome by its probability, and adding all of these numbers up. This gives us a long-run average value for our random process.
Expected value is helpful for assessing gambling outcomes. If my expected value for playing the game, based on the cost of playing and the probabilities of winning different prizes, is positive, then in the long run the game will make me money. If expected value is negative, then this game is a net loser for me.
Powerball and similar lotteries are a wonderful example of this kind of random process. As of October in Powerball, five white balls are drawn from a drum with 69 balls, and one red ball is drawn from a drum with 26 balls. (As an aside, that rule change is why prizes can get as big as the current record jackpot: The probability of winning the jackpot is much lower than it used to be.)
Prizes are then given out based on how many of a player’s chosen numbers match the numbers written on the balls. Match all five white balls and the red Powerball, and you win the jackpot. In addition, several smaller prizes are won for matching some subset of the drawn numbers.
Powerball’s website helpfully provides a list of the odds and prizes for each of the possible outcomes. We can use those probabilities and prize sizes to evaluate the expected value of a $2 Powerball ticket. Take each prize, subtract the price of our ticket, multiply the net return by the probability of winning, and add all those values up to get our expected value:
At first glance, we seem to have a positive expected value at $3.45. The situation, however, is more complicated.
Annuity versus lump sum
Our first problem is that the headline $1.5 billion grand prize is paid out as an annuity: Rather than getting the whole amount all at once, you get the $1.5 billion spread out in smaller — but still multimillion-dollar — annual payments over 30 years. If you choose to take the entire cash prize at one time instead, you get much less money up front: The cash-payout value at the time of writing is $930 million.
Looking at the lump sum, our expected value drops dramatically to just $1.50:
The question of whether to take the annuity or the cash is somewhat nuanced. Powerball points out on its FAQ site that in the case of the annuity, the state lottery commission invests the cash sum tax-free, and you pay taxes only as you receive your annual payments, whereas with the cash payment, you have to pay the entirety of taxes all at once.
On the other hand, the state is investing the cash somewhat conservatively, in a mix of various US government and agency securities. It’s quite possible, though risky, to get a larger return on the cash sum if it’s invested wisely. Further, having more money today is frequently better than taking in money over a long period of time, since a larger investment today will accumulate compound interest more quickly than smaller investments made over time. This is referred to as the time value of money.
Taxes make things much worse
As mentioned above, there’s the important caveat of taxes. While state income taxes vary, it’s possible that combined state, federal, and in some jurisdictions local taxes could take as much as half of the money. Factoring this in, if we’re taking home only half of our potential prizes, with the headline annuity payout our expected value drops to $0.73:
Of course, looking at the lump sum is probably a more accurate description of our prize. After taxes are taken out against that, we end up with a negative expected value of -$0.25, making our Powerball ticket a bad investment:
Splitting the jackpot
Of course, all the above analysis assumes that you and you alone win the jackpot. However, with insanely high jackpots come insanely high ticket sales. According to Lottoreport.com, a site that tracks lottery sales, over 440 million Powerball tickets were sold before Saturday’s drawing. This is by far the largest number of Powerball tickets sold for a single drawing going back at least to 2012, when Lottoreport’s figures begin.
With this many tickets out there, the likelihood that not only one person, but multiple people, could hit the magic numbers and split the jackpot gets to be pretty high. We can model the probabilities of the number of winners for a given number of tickets sold using what’s called a binomial distribution, a very handy tool in probability theory and statistics:
If 500 million tickets are sold — a not unreasonable possibility given the 440 million sold for Saturday’s drawing — it’s actually more likely that two or more people will split the jackpot than having just one winner or another round of no winners.
This has a pretty serious effect on our expected value calculations. We no longer just have a tiny probability of winning $1.5 billion; we have a smaller probability of that, as well as a tiny probability of winning just half of the jackpot, as well as a third, and so on. Further, those new tiny probabilities are effected by how many tickets are sold, which we won’t know for sure until after the drawing.
Factoring that in, here’s what happens to the expected value of the pre-tax prizes at different levels of tickets sold. If more than 424 million tickets are sold, the lump sum becomes a losing bet. If 843 million tickets are sold, the annuity’s expected value goes negative:
Factoring in taxes makes the game look even worse. As we saw above, the lump sum always has a negative expected value after taxes, even if we win the entire pot and don’t have to share. After taxes, the annuity becomes unprofitable if more than 207 million tickets are sold, a number that seems extremely likely given the mania around the game:
So, even though Wednesday night’s jackpot is insanely high, when we consider the extremely low likelihood of winning, taxes, and the possibility of splitting the jackpot, buying a Powerball ticket is probably not a good investment.
Good luck to everyone playing anyway!
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