## 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.

Greg Grothaus said...

To update for tonight's \$464M jackpot, the financial decision is only good if you have at least \$286,655,926. Still to rich for my blood.

Greg Grothaus said...

A friend of mine also pointed me to this which takes a look at the expected value considering the number of players: http://www.circlemud.org/~jelson/megamillions/

The graph at the bottom is the most interesting.

Flash Gordon said...

"A tax on people who aren't very good at Math", funny as Greg. Interesting post, nice work.

Mark said...

I feel this is a good point you make as well as very controversial. Despite the fact some of what you say is true, people just don't look at it like that, they all believe somewhere in their hearts there is little hope of the big win coming to them so why not take the gamble? I think its well worth expanding upon this point, including some of the multiple views people have towards the lottery or Mega Millions.

Greg Grothaus said...

Mark, there is something to say for the fact that some of the value of buying a lottery ticket is reward to the purchaser in terms of excitement, hope, entertainment, etc.

If someone with a large bankroll buys a \$1 ticket, the risk is pretty low and the entertainment value may exceed \$1 even if there is a loss.

If someone goes and blows their paycheck on tickets, it's a different story.

Mark said...

Greg, I will touch upon the point of the person who blows his whole paycheck in hope to win Mega Millions. I feel this type of a person is an addict and has no control over his or her finances.

Some years ago I was contracted to a company in my home city London UK to implement a large network of diskless thin clients, there I joined a syndicate with several of the workers in hope we could win Mega Millions. There were around 20 members in this syndicate, we marked the lottery twice a week, each member at the time paid £1.00 a ticket, from time to time we would win the occasional £10.00 then I persuaded the syndicate to use the lottery wheeling system and we were winning at least £40 every 2 to 4 weeks and at one point we got 4 numbers and won £1500 however some members started to get angry with my lottery wheeling system, as a result it was discontinued. This was over ten years ago.

Recently upon a trip back home to London I stumbled upon some of the workers who still mark the syndicate in hope to win Mega Millions, they told me they have never had a bigger win since parting from the lottery wheeling system which I introduced to them in 2001. Two members were lottery addicts and frequently used their whole paycheck to purchase lottery tickets in hope of winning Mega Millions.

I feel the reason why people do such things can be connected to many reasons similar to a smoker, drinker, or any other addicts; in addition to that I think they feel in the back of their mind the chance of the big win is there.

Tony said...

It could be you... ;)