Maximize Your Chances of Winning the Lottery

Posted by Dan Hung

August 25, 2009

Just a little digression tonight, the Megamillions jackpot is up to $252 million tonight and I thought it’d be fun to do a little quick analysis of your chances of winning. For those that don’t know, the game is played by choosing five numbers from a set of 1 to 56 (without reptition) and then a “megaball” number between 1 and 46.

So, for those that know a little about probability, there are (56C5)*46=175,711,536 different permutations of numbers that you can choose. Rather than go through all the probabilities of winning, the lottery actually publishes its own listing of probabilities to help you out. Check it out below:

Megamillions Probabilities

So, knowing these probabilities, let’s calculate expected values. Expected values are calculated simply by multiplying the expected result by the probability of it happening. Here’s how we would do it for playing the Megamillions.

Subtract $1 from all the payouts listed in the above table (the cost of playing)
Multiply each payout by the probability of it happening and sum them
Calculate the probability of not winning anything (1-sum of the probabilities of winning)
Multiply the probability of not winning by -$1 (the cost of playing)
Add this to the result from Step 2

At a $252 million jackpot, the expected value of buying a lottery ticket is $0.61. That means that if you spend $1 to buy a ticket, you can expect to see a return of 61 cents, or less than you “invested” in the game. Probabilistically, it’s entirely not worth it to play.

What would the jackpot have to be to make it “worthwhile” to play? Backing into an expected value of $1 for playing yields a target jackpot of $320 million.

Obviously, the above scenarios ignore two important facts. First, you face taxes on any money you win (except winnings <$600 which you can get in cash at your local convenience store and probably not report… not that I advocate not paying your taxes…). Second, the “lump sum” jackpot is usually approximatly 40% less than the headline jackpot number. So, the real “breakeven” expectation jackpot is more like $904 million!

So, let’s bring this back to some practical lessons for investing. First and foremost, “rational” investors should never play a game that has an expectation below the amount you pay to play. Yet, somehow, the record Megamillions jackpot is $365 million. Thus, it would seem there’s a large amount of people out there willing to play the game well before any rational expectation of winning, let alone a miniscule probability of winning the jackpot. If so many are willing to spend even a dollar on the lottery, how are we so sure that capital markets are efficient!?

Arguments against popularly held market theories aside, it’s clear that people are irrationally willing to play games of chance. If this is true, then might this point us to an interesting investment thesis? Why not invest in purveyors of games of chance? Casinos and gaming is a massive industry and there are many publicly traded options available. Publicly traded casino operators include MGM Mirage (MGM), Wynn Resorts (WYNN), and Las Vegas Sands (LVS) all trade publicly. Some racetrack and slot machine operators are publicly traded as well like Churchill Downs (CHDN), Penn National Gaming (PENN), and Empire Resorts(NYNY). Then, there are the Companies which supply these gaming operators – International Game Technologies (IGT), Bally Technologies (BYI), and WMS Industries (WMS).

Or, you could just go after investments in all things vice – gaming, tobacco, and firearms – as described in my article from February of last year – Indulge Your Vices.

Full disclosure: No positions in any of the stocks mentioned in this post.