Who wants to win the lottery?

So everyone would love to win the lottery right? Just think of what you could do if you had even $1 million dollars to spend. You could buy a dozen tacos a day at Taco Bell for the rest of your life. And your children’s lives. And their children’s lives. 228 years to be more precise. Or you could pay to send the entire family from Cheaper by the Dozen to the average state university–even if they each took an additional two years to graduate. And that’s just for $1 million. Payouts are usually much higher than that…

So what’s the catch?

(If you came here just for the Powerball simulation, it’s down at the bottom of the page. Click here to go straight there.)

Glad you asked. It turns out that the catch is pretty straight forward. All you need to know is a little statistics. Or you could just go to the Powerball website (for example) and see what odds they’ve calculated for you. The chart below has the rough odds either for the standard $2 ticket or for the $5 PowerPlay ticket at $4 with tweaked payouts.

MatchingPowerball?Prize ($2 ticket)Prize ($4 ticket)OddsOdds
5YesJACKPOT!JACKPOT!1 in 175 million
5No$1 million$2 million1 in 5 million
4Yes$10 thousand$40 thousand1 in 648 thousand
4No$100$2001 in 19 thousand
3Yes$100$2001 in 12 thousand
3No$7$141 in 360
2Yes$7$141 in 706
1Yes$4$121 in 110
0Yes$4$121 in 55

But what does that actually tell you? Well, quite a lot actually. If you take the odds of each chance of winning and multiply it by the payoff (less $2/$4 for the ticket), that will give you the expected value for that payoff. For example, if you only consider the Powerball, the expected value would be:

(1/35)(2) + (34/35)(-2) \approx -1.88

So every time you played (if only the Powerball counted) you’d be out an average of $1.88. You may win sometimes (1/35 of the time even), but overall, you’d lose more than you won. What would the payoff actually have to be for it to be a “fair” game–meaning that over time you’d neither make more or lose it?

(1/35)(x-2) + (34/35)(-2) \approx 0
x \approx 70

(Which really makes sense, as you have a flat 1/35 chance of winning and are paying $2 per game.)

But you might think that the other payoffs will even the odds out right? After all, they’re all significantly higher payouts. Especially that Jackpot. What if we just took that? Well, taking the current Jackpot of $149 million:

(1/175000000)(149000000) + (174999999/175000000)(-2) \approx -1.15

Better actually. But you’re still out $1.15 on average per game. Really, the Jackpot has to be over $350 million for the expected value to actually be positive. What’s interesting is that sometimes does actually happen. You actually do get people that will go out and buy a lot of tickets around that time, thinking that they’re bound to make it big. But there’s one more wrinkle to consider… What happens if two people both guess the correct numbers? Well, it turns out that they’d split the jackpot. So instead of that $350 million payout, you’d only get half that. And then there are taxes (quite a lot of them actually–the federal tax alone is about 25% on such winnings).

I guess my point is that trying to get rich by playing the lottery doesn’t really make sense. With the sheer numbers of people playing, you’re almost certainly going to get some winner–but the odds of it actually being you? Not so good.

But what about the game in general? What is the overall expected value of a Powerball ticket? For that, we have almost everything we need in that chart up there. But what we still need is the chance that you get none of the ones above at all. Luckily, it’s easy enough to calculate, just sum up all of the odds above and subtract from one. That ends up giving us… 96.8% or approximately 1 in 1.03. So it’s really good odds that you’re just going to flat out lose the $2 you put into a ticket. But, now that we have that, we can work out the full expected value of a ticket:

Assuming today’s estimated Jackpot of $149 million, we have:

1/175223510 * (149000000 - 2)
1/5153632.65 * (1000000 - 2) +
1/648975.96 * (10000 - 2) +
1/19087.53 * (100 - 2) +
1/12244.83 * (100 - 2) +
1/360.14 * (7 - 2) +
1/706.43 * (7 - 2) +
1/110.81 * (4 - 2) +
1/55.41 * (4 - 2) +
1/1.03 * (-2) = -\$0.79

So every time you buy a Powerball ticket, your going to lose an average of 80 cents.

But what’s really interesting is if you play the Powerplay. The odds don’t actually change, all that changes is that you’re paying twice as much and that all of the non-Jackpot prizes go up. So what’s the expected value now?

1/175223510 * (149000000 - 4) +
1/5153632.65 * (2000000 - 4) +
1/648975.96 * (40000 - 4) +
1/19087.53 * (200 - 4) +
1/12244.83 * (200 - 4) +
1/360.14 * (14 - 4) +
1/706.43 * (14 - 4) +
1/110.81 * (12 - 4) +
1/55.41 * (12 - 4) +
1/1.03 * (-4) = -\$2.30

Dang. So by buying a Powerplay ticket, you’re actually losing about three times as much. Really though, this does make sense. For the expected value to remain the same with twice the pay-in, you’d have to double all of the prizes. And all of the prizes did double… except for the Jackpot. And that $149 million has a bit of weight all to itself.

What about if the Jackpot is different? Well here’s a little tool that can help you calculate it:

Normal EV:
Powerplay EV:

But you don’t have to just believe the math. You can try it out for yourself. I’ve written a simple script below that simulates a series of Powerball drawings. Go ahead and fill out the chart below to try it out. Since the simulation is based on Powerball, you’ll need to enter 5 different numbers in the first box, each from 1 through 59. The second box will be your Powerball and should be in the range 1-35. Finally, enter how many tickets you want to buy in the third box (1-100). If you’d rather just choose some random numbers, there’s a button for that too.

Feel free to play as long as you want, it doesn’t cost you anything. And if anyone actually does get a Jackpot, make sure to let me know. 😄

Your numbers:
Times to play:

If you’d like, you can download the source here: lottery source