Photo: MassStateLottery via YouTube
The Powerball lottery has now hit $500,000,000, and one thing is now certain: It’s actually statistically justifiable to buy in. We’re looking at expected value now.
In essence, expected value looks at how much you’re expected to earn by playing a game of chance.
Conveniently, Powerball publishes their probabilities so we can easily calculate the chance.
Here’s how you calculate the expected value of an event.
Let’s say you have four dollar bills of different denominations in a bag. You can’t see them, but I tell you that one is a 20, two are singles, and one is a five. I’m charging you six bucks to reach in and grab a bank note. Should you play?
Well, the probability of drawing any of the four bills is 0.25. There are two singles, so the probability of drawing a single is 0.5. If we multiply the probability and payout of each event, and sum that for all events, we get the expected value of playing the game.
Here’s the expected value for the money game. Keep in mind I’m charging you $6 to play:
EV = ($20—$6)*(0.25) + ($5—$6)*(0.25) + ($1—$6)*(0.5)
EV = ($14*0.25) + (-$1*0.25) + (-$5*0.5) = $0.75
So, on average, you’ll make 70 five cents by playing the game. So you should, theoretically, play.
It’s the same way with the Powerball lottery now. Powerball costs $2 to play. You win the half-billion by getting each of the five numbers correct (of 59 possible numbers) plus the one “Powerball” correct (of 35 possible numbers). The probability of doing this is 1 in 175,223,510. But, there are a number of other prizes for getting four of five numbers right and so on.
As a result we can make this table. The key here comes in the last column, which is the amount of money gained or lost times the probability of that event:
The net amount of money you’re expected to gain from paying in powerball tomorrow is $1.21. This already accounts for the two dollar buy in.
Keep in mind that this is only buoyed by the outrageously high jackpot. 97% of people will lose $2 in the course of play. But it’s that one in two hundred million getting a half-billion that drags this number up.
In fact, Powerball is usually a loser’s game. It usually has a negative expected value.
The break even jackpot point for Powerball is at $276,505,966. Anything below that jackpot has a negative expected value to play for $2. Anything above that has a positive expected value.
Keep in mind that there are a couple of caveats here. We assumed you’re not doing the PowerPlay multiplier, because that exempts you from the Jackpot.
Also, it’s always possible that someone else could win and you’ve have to split the winnings. That angle severely complicates the calculation overall.
Either way you look at it, though, there’s an unusual statistical event happening on Powerball this week — a lottery with a positive expected value — and it’s not going to last forever.
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