The US Powerball Mega Millions jackpot for Wednesday’s drawing, after no one won on Saturday, is up to $US430 million as of 12:30 PM ET Tuesday.
That is a pretty huge chunk of money. However, as we saw with last week’s similarly large Mega Millions jackpot, which someone did win on Saturday, taking a closer look at the underlying maths of the lottery shows that it’s probably a bad idea to buy a ticket.
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.
Lotteries are a great example of this kind of probabilistic process. In Powerball, for each $US2 ticket you buy, you choose five numbers between 1 and 69 (represented by white balls in the drawing) and one number between 1 and 26 (the red “powerball” in the drawing). Prizes are then given out based on how many of the player’s numbers match the numbers chosen in the drawing.
Match all five of the white balls between 1 and 69, and the extra red powerball number between 1 and 26, and you win the jackpot. After that, smaller prizes are given out for matching some subset of those numbers.
The Powerballs 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 $US2 ticket. Take each prize, subtract the price of our ticket, multiply the net return by the probability of winning, and add all those values up to get our expected value:
Already, we can see that a Powerball ticket isn’t a great investment: We end up with an expected value of -$US0.21, making a ticket a losing proposition.
Unfortunately, when we consider some other aspects of the lottery, it gets far worse.
Annuity versus lump sum
Looking at just the headline prize is a vast oversimplification.
First, the headline $US430 million grand prize is paid out as an annuity, meaning that rather than getting the whole amount all at once, you get the $US430 million spread out in smaller — but still multimillion-dollar — annual payments over 30 years. If you choose instead to take the entire cash prize at one time, you get much less money up front: The cash payout value at the time of writing is $US273.4 million.
Looking at the lump sum, our negative expected value gets even worse, falling to -$US0.74:
The question of whether to take the annuity or the cash is somewhat nuanced. The Powerball website says the annuity option’s payments increase by 5% each year, presumably keeping up with and somewhat exceeding inflation.
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, although 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
In addition to comparing the annuity with the lump sum, there’s also the big 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, our expected-value calculations move deeper into negative territory, making our Powerball investment an increasingly bad idea. Here’s what we get from taking the annuity, after factoring in our estimated 50% in taxes. The new expected value is now underwater, at -$US0.94:
The hit to taking the one-time lump sum prize is just as devastating:
A Powerball ticket, then, is a pretty poor “investment.”