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!
I'm here to debunk a silly theory we've seen floating around the internet - that goes like this... The 5/18/2013 Powerball drawing will be $600,000,000, and odds to win are 1/175,000,000 - so each $2 ticket is worth $3.42!
And we'll show you how using the somewhat simple math. The people at Powerball are smart - smart enough they aren't going to run a lottery expecting to lose money.
We previously discussed the expected value of the record shattering Mega Millions drawing (negative), then built a calculator so you can do the math on arbitrary drawings. Recently we talked about how people have beat the lottery - but, not surprisingly, the proven cases generally came from smaller state level lotteries... not country-wide lotteries vetted by tons of auditors and actuaries.
Buying Powerball Tickets For This Drawing is Like Throwing Away at Least 50 Cents
We'll get to our math in a second - but, sorry to say, there isn't a positive expected value on Powerball tickets, no matter what news organizations are implying.
Long story short:
Your expected value is around $1.46.
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:
|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.
Sorry to Burst Your Bubble
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!