Should You Play Every PowerBall Number? (If you could)

The Power Ball has reached a $340 million dollar jackpot, and a lot of people are talking about it in the news.

I got to thinking, how big does the jackpot have to get before it becomes statistically profitable to buy a ticket for EVERY SINGLE NUMBER. (positive expected value, or Plus EV, to us stat geeks).

So, here are all the calculations I did. For those of you uninterested in the nitty-gritty, go all the way to the bottom for the findings.

$1 gets you a 1 in 150 million chance of winning.
The jackpot is $340 million.

If you play every number, it would then cost $150 million dollars.

If you play every number, and nobody else wins the jackpot you win $340 million.
340 - 150 = $190 million PROFIT.

If the jackpot is split with one other winner, you’d win $170 million.
170 – 150 = $20 million PROFIT.

If the jackpot is split with two other tickets, you’d win $113 million.
113 – 150 = $37 million LOSS.

The post-gazette stated yesterday that there is about a 55% chance someone will win tomorrow.

So, we can assume that if you play EVERY ticket, there is a 45% chance you’d be the only winner, and a 55% chance that you’d have to split the pot.

If we make some conjecture here, and say that the chance of only 1 other winner is 50%, a 30% chance of 2 other winners, and a 20% chance of 3 other winners, we can come up with an approximate expected value for playing every number.

So, $190 * .45 + $20 * (.55) * (.5) + (-37) * (.55) * (.3) + (-65) * (.55) * (.2) = 85.5 + 5.5 – 6.105 – 7.15

Or $77.745 million profit.

But, of course, you need to pay "the man". If tax is 35%, does that take away your advantage?

The EV equation becomes 74 * .45 + (-36.5) * .275 + (-73.55) * .165 + (-91.75) * .11 = 33.3 – 10 – 12.1 – 10

(To be more fair, I added a +3 to all those numbers, because we really only need to pay for 147 million numbers).

-- RESULTS --

Or about $1.2 million profit for playing every number. Damn, that’s still profitable, but just barely, considering you invested $150 million. You'd have a 45% chance of winning a lot of money, and 55% chance of losing money, which probably plateaus as high as a $100 million loss.

I don't recommend buying every ticket.

If there were no taxes, it would be completely different. In that case you could probably start a mutual fund just to buy lottery tickets and give away the dividends to shareholders, over time it would certainly have a net gain better than the stock market. *grin*

Also, as the jackpot continues to grow, the chance you'd have to split the pot also grows because more people play when the jackpot is high. So I'd speculate that the statistical profitability of the game is never going to make it worth buying every ticket.