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The Powerball jackpot for Wednesday's drawing, after no one won on Saturday, is up to $650 million as of 11:30 a.m. ET Monday.

That is a pretty huge chunk of money. However,as we saw before Saturday's drawing, when the jackpot was $535 million, taking a closer look at the underlying math 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 isexpected value.

The expected value of a randomly decided process is found by taking all the possible outcomes of the process, multiplying each outcome by its probability, and adding all those 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. InPowerball, for each $2 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"). Prizes arebased onhow many of the player's chosen numbers match the numbers drawn.

Match all five of the numbers on the white balls and the one on the red powerball, and you win the jackpot. After that, smaller prizes are given out for matching some subset of the numbers.

The Powerball website helpfully providesa list of the odds and prizesfor each of the possible outcomes. We can use those probabilities and prize sizes to evaluate the expected value of a $2 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:

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At first glance, we end up with a positive expected value of $0.54. That seems as though it might make sense to buy a ticket, but considering other aspects of the lottery makes things much worse.

Annuity versus lump sum

Looking at just the headline prize is a vast oversimplification.

First, the $650 million jackpot is paid out as an annuity, meaning that rather than getting the whole amount all at once, it's 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 $411.7 million.

If we take the lump sum, then, we end up seeing that the expected value of a ticket drops below zero, to -$0.27, suggesting that a ticket for the lump sum is a bad deal:

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The question of whether to take the annuity or the cash is somewhat nuanced. ThePowerball websitesays 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 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, since a larger investment today will accumulate compound interest more quickly than smaller investments made over time. This is referred to as thetime value of money.