## Mar 24, 2012

### Mega Millions

I for one think the lottery is a tax on people who aren't very good at math.  If I wanted to gamble, I'd buy a casino.

Mega Millions is the big multi-state lottery that has in recent days grown it's jackpot to the point where the payout could be worth the risk.  The cash payout is currently at \$255 million (the \$356M number is an annuity) and growing.  To win, you buy a \$1 ticket where you must pick 5 random numbers correctly out of a pool of 56 and 1 random number correctly out of a pool of 46.  The odds of a correct pick are 1 in 56C5 x 46 or 1 in 175,711,536.

Is this a "good bet"?

For the sake of simplicity, let's ignore taxes as well as the possibility that there is more than one winner (thus splitting the payout).  Also, most people's utility for money is non-linear (ie: beyond a certain point, more money doesn't matter as much any more).  Those are important issues to consider in real life as they have a fairly large effect.

To estimate the expected net returns, P(win) x jackpot - cost =
=

=     \$0.451.

45% return on your dollar in just a few days sounds like a great investment.  Mortgage your house, max out your credit lines, sell your stocks, and invest in tickets!

The problem with bets is that even if you have the odds in your favor, you can still lose everything.  How do you choose what to bet then?  Bet too little on a good gamble and you'll leave money on the table.  Bet too much and the losses will keep wiping too much of your winnings out.  It turns out that there is formula that predicts the optimal size in a series of bets which will maximize your winnings in the long run.  This is called the Kelly criterion.  It determines a bet size based on your odds and your current bankroll available to wager.  As your bankroll grows, you bet more.

Let's say that you were only allowed to buy one mega-millions ticket per round with the 1 to 175,711,536 odds of winning \$255M.  Is this one ticket a sound investment?  The Kelly criterion tells you the fraction of your bankroll you should invest in this bet.  The formula is simple enough, divide the expected net returns (\$0.451) by the net winnings if you win (\$255M).  The result is 1.77x10-9 or 1 in \$565,410,199.

If and only if you have at least \$565,410,199 to invest, buying a \$1 ticket is a mathematically sound investment.

Buying two tickets in the same lottery round is a slightly different gamble than buying 1 ticket each round.  It's a little better odds.  Taken to the extreme, buying 175,711,536 tickets guarantees a win whereas buying 1 ticket in each of 175,711,536 rounds does not.  A modified Kelley criterion can evaluate the case where you buy multiple tickets too, and it's going to be more favorable.  Unfortunately, I don't have the time at the moment to add that to the post.  If there is interest, perhaps I'll return and see if I can work through that math.