When playing the lottery appears to makes sense

Sophisticates commonly claim that playing the lottery is foolish.  “A tax on people who don’t understand math” is one way I’ve heard it put.

Usually this is true.  However, consider the upcoming Mega Millions game.  The odds of winning are 1 in 176 million.  However, the payout if you win is estimated at $325 million.  This means that the expected value (naively calculated) is 325/176 = 1.85.  In other words, the average investment in a lottery ticket will return 85% in merely days.  You are unlikely to find a surer bet anywhere in the financial world.

So I encourage all my readers to rush out and buy lottery tickets.  In fact, following my own advice, I’ve bought 10 tickets and can therefore expect to win $18.50 on Friday!

The odds are that this dude didn't win the mega millions.

Now, my more astute readers may have noticed that I parenthetically mentioned that my expected value calculation was naive.  Unfortunately, a more careful analysis reveals that the expected value of a $1 ticket must take into account that the $325 million is paid out over 26 years.

To be comparable, then, the cost of the ticket should be calculated as the time value of $1 over 26 years — which would be… well, depending on the interest rate it could be anything.  Very likely it will be a lot more than $1, though.

The easier alternative is to compare the $1 ticket to the amount of money that could be accepted in a lump sum immediately, which is roughly $206 million.  It still appears that the ticket has a positive expected value, but, alas, we must also account for the fact that as the jackpot grows in size, more people play the game and therefore the odds of multiple winning tickets goes up — which means a split jackpot.

I won’t be able to come up with exact numbers without going through boring calculations, but if you are interested, check out this paper by John Corbett and Charles Geyer.  The long and the short of it is that you are unlikely to come out ahead.

And then, of course, there’s the tax bill to consider…