The Saturday Powerball Drawing: You Do Not Have a Positive Expected Value!

May 17th, 2013 by
PK

Look! A sucker!

Ed: Whoops, forgot 5/5 with a missed Powerball in the subprizes.  Shows what I know about how the Powerball works, eh?  It's still a bad idea.  Numbers Revised!

Hey everyone, it's your favorite lottery killjoy PK!

Your ticket will cost $2. That's a negative expected value. And, oh yeah - you'll be taxed on your winnings - so that$1.46 might be $.80 or less! Buy some S&P 500 Index Funds... trust me (here's a historical calculator!). Why is all the math wrong? It all boils down to the number of people who will win. If more than one person wins, the prize will be split equally amongst the winners. However, the odds are actually much greater that either 0 or more than one person will win. Here's what we calculated:  Winners Probability 0 18.0504% 1 30.9024% 2 26.4524% 3 15.0955% 4 6.4609% 5 2.2122% 6 0.6312% 7 0.1544% 8 0.0330% 9 0.0063% 10 0.0011% The Boring Math Only 10% Of You Will Read How did we get those? We did it the same way as we did for the Powerball, but this time we'll try to step through it quickly with our assumption of ticket sales. The only jackpot approaching today's$600 Million (actually, $376,900,000 Cash Prize - remember, it's an annuity otherwise) was the 11/28/2012 drawing with the$587 Million nominal prize.  It sold 281,565,987 tickets.

Let's assume 300,000,000 are sold.

Reproducing the distribution is easy, especially in a spreadsheet program!  Your Poisson number is 300,000,000 tickets sold * odds of winning (1/175223510), or 1.7121.  Now, simply take the odds of each (as shown above!), and multiply by what the jackpot would be in that situation for a single ticket.  So, 0 winners = $0, 1 winner =$376,900,000 * 30.9%, 2 winners = $188,450,000 * 26.5%, etc. If you sum the expected prizes, you get$193,539,183.00... that's the expected jackpot, folks.  That's only $1.10 per ticket. The smaller prizes are simple to calculate - they don't split (and I'm not attempting to factor Power Play - this is my quick estimate folks; my sister is here, haha). Just take straight odds, and you'll get this table:  Description Prize Odds Powerball only$4.00 0.0180472839 1 number plus PB $4.00 0.0090244563 2 numbers plus PB$7.00 0.0014155684 3 numbers; no PB $7.00 0.002776698 3 numbers plus PB$100.00 8.16671199191822E-005 4 numbers; no PB $100.00 5.23902254508572E-005 4 numbers plus PB$10,000.00 1.54088912630909E-006 5 numbers; no PB $1,000,000.00 0.000000194 Add it up - around 17 cents of expected value. Sum it up, and you've got$1.46 in expected value.

Yeah, my bad.  Sorry to be such a curmudgeon... but if you all send me $2, I'm sure we could work out a deal where I send you each$1.46.  Bernie Madoff has nothing on me!

PK

PK started DQYDJ in 2009 to research and discuss finance and investing and help answer financial questions. He's expanded DQYDJ to build visualizations, calculators, and interactive tools.

PK is in his mid-30s and lives in New Hampshire with his wife, kids, and dog.