By Marlon E from USA (BINGO!) [CC BY-SA 2.0 (http://creativecommons.org/licenses/by-sa/2.0)], via Wikimedia Commons

The Powerball jackpot is $1.4 billion. Is it worth it to play?

Short answer: No.

Long answer: Before you consider the lottery and all its nuance, consider a game you can play at home. Suppose your friend lets you flip a coin for $5. If it’s heads, he’ll pay you $11. If it’s tails, you get nothing. Is this game worth playing?

Let’s look at the expected return, which is the sum of possible returns for each outcome weighted by that outcome’s probability (i.e. the probability of an outcome multiplied by its return). For this game, the expected return is

(0.5)*($0) + (0.5)*($11) = $5.5

Since your expected return is greater than the cost per play, you can expect to make money playing this game. Now imagine your friend allows you to bet a small amount that a very specific sequence of coin-flips or dice-rolls will occur, and if it does occur, you get a big payout. In essence, a lottery. The same math applies if you know the likelihood (or unlikelihood) of your specific sequence, the cost per play, and the award for winning.

 

While the Powerball jackpot is $1.4 billion, they say that it’s really a $868 million cash value if you take it in one payment [1]. After taxes, which amount to nearly 40%, you’d be left with close to $500 million.

A Powerball ticket consists of 5 numbers chosen from 1 to 69, and an additional number chosen between 1 and 26, so there are (69 choose 5)*26 = 292,201,338 possible tickets. Thus, we can calculate our expected return:

(1/292,201,338)*($500 million) = $1.71

If a ticket is $2, you’re losing money. If you spent $584 million to buy tickets with every possible lottery number so that you could guarantee a win, you’re really only guaranteeing that you lose $84 million. (In fact, some people do this for scratch-off lotteries whose expected returns change depending on the size of the jackpot. Once the pot gets large enough the expected return exceeds the cost per ticket, and with some wealthy donors and some determined scratchers you can make a healthy profit [2]. The best part? It’s totally legal.)

 

The point? The odds are so overwhelmingly against you that I wouldn’t recommend wasting your time on buying a ticket. If you look at odds of 1:292,201,338 and think “there’s a chance” then consider one last thing. The National Highway Traffic Safety Administration tracks car accident fatalities in the US. In 2013 it was close to 1.09 fatalities per 100 million miles traveled [3]. This means you are more likely to die on the drive to go buy a lottery ticket than you are to buy the winning ticket.

Save a life. Save $2. Don’t buy a lottery ticket.

 


 

 

image credit: Wikimedia Commons

 


 

 

Have a question? Send it to matt@quarksandcoffee.com