The Powerball lottery drawing for Saturday evening has hit a record high estimated jackpot prize of $800 million as of this writing, after no one won the jackpot in Wednesday’s drawing.

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 2015 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 new record: 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, there are several smaller prizes 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 a first glance, it looks like we have a positive expected value at $1.06. However, the situation is more complicated.

## Annuity vs. lump sum

Our first problem is that the headline $800 million grand prize is paid out as an annuity: Rather than getting the whole amount all at once, you get the $800 million spread out in smaller — but still multimillion-dollar — annual payments over the course of 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 $496 million.

Looking at the lump sum, our expected value drops dramatically to just $0.02:

The question of whether to take the annuity or the cash is somewhat nuanced. Powerball points out on their FAQ site that in the case of the annuity, the state lottery commission invests the cash sum tax-free, and you only pay taxes 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, 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

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 only taking home half of our potential prizes, with the headline annuity payout we have a negative expected value of -$0.47, making our Powerball “investment” a bad idea:

The hit to halving the cash one-time prize is equally devastating:

When you consider issues like taxes and their effect on the expected value of a Powerball ticket, you can see that the lottery is a pretty bad “investment.”

## So when does it make sense to buy a ticket?

Well, probably never. But for fun, let’s consider when we’d actually get a breakeven expected value of zero or higher on our “investment”.

We need the expected value contribution of the jackpot to counter balance the very negative and very likely outcome that we win nothing, offset by the contributions from the smaller prizes. Adding up all the non-jackpot prizes and our highly likely outcome of just losing our $2, we need the jackpot expected value contribution to be about $1.68:

So, we need to solve the equation: Jackpot x (1 in 292,201,338) = $1.68. Divide both sides by that insanely low probability that we win the jackpot and we get our desired take home winnings to be $490,933,821.72.

Under our assumption that we could lose up to 50% of our winnings to taxes, that means we need the pre-tax prize to be $981,867,643.44.

Since we are inclined to take the lump sum rather than the annuity, factoring in the importance of time value of money, if we assume the same ratio between the annuity and lump sum that currently holds — a $800 million headline prize compared to a $496 million cash lump sum prize — that means that, to even get close to a breakeven value, we would need the headline annuity prize to be a whopping **$1,583,657,489.42**.

Of course, even at that astonishingly high nearly $1.6 billion headline prize, we could run into the problem of the possibility of splitting that prize with another winner. Factoring in further issues with the lottery then, we would need a prize at least twice as high as the current record high to even consider buying a ticket, and even then it’s still probably a bad idea.

Good luck to everyone playing!

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