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Binomial distribution and Mega-Millions Lottery ROI

As of this morning, the Mega-Millions lottery for this week is at a record-setting grand prize level of \$540M (when taken in installments) or \$389M lump sum payout.   The odds of any given ticket winning is 1 in 175,711,536.  So, if you are a risk-neutral "investor" and you buy a ticket,  your mean return-on-investment (ROI) may actually be positive in this case -- a highly unusual situation for a lottery.  Hopefully you see the main opportunity that this presents: the opportunity to learn about the Binomial distribution!

If it were as simple as saying that you had a 1 in 176M chance of winning \$389M, then your mean winnings would be \$2.17 for a \$1 ticket.  That would be a 117% ROI in only a few days time.  Of course, it is not that simple due to two complicating factors: (the potential for) multiple winners, and taxes.  When there are multiple winning numbers, the prize is split between winners, so even if you win you might receive only half, or one-third of the total payout.  And of course, your take-home will be greatly reduced by taxes on the winnings.

The Binomial distribution provides the important component for modeling the number of simultaneous winners.  `Binomial(n,p)` takes two parameters: n = the number of tickets, and p = the probability of any one ticket being a winner.  The result is an integer-valued discrete distribution giving the number winners.   In the lottery example, we know p, so the critical parameter for estimating ROI is n, the number of tickets sold.  Try it out:  In Analytica, create a chance node and give it the definition:  `Binomial( 100M, 1/176M)` and view the Probability Mass view.  The graph shows a 57% chance of zero winners, 32% of a single winner, 9% of two winners, and 1.7% of three winners, 0.3% chance of 4 winners for this example in which 100M tickets are in play.

Today's San Jose Mercury News reports that as of yesterday, 50M tickets had been sold for this week's drawing.  That's already on the high side for any lottery drawing, but since this is the largest jackpot in history, that number is certain to increase further.   This number is of course, the primary source of uncertainty involved here.

If you win, your take home prize is:  Take_home := `Payout / Num_winners * (1 - Tax_rate)`.   Num_winners is defined as `Binomial( tickets_sold, odds)`.

Your return on investment is:  ROI := `Take_home * odds - 1`.

Your tax rate depends on where you live, but will likely consist of Federal income tax (25%), state income tax, and perhaps other medicare and Medicaid taxes.  The ROI calculation is very sensitive to the precise tax rate, so if you want to treat this as more than a fun diversion, you may need go look up the precise numbers.  You can plug in your own tax rate into the attached model if you'd like to explore further.  The graph below shows a line for tax rates of 0%, 25% and 49%, as a function of the number of tickets sold (which is the big unknown).

I offer this diversion as a nice opportunity to learn about the Binomial distribution.  I have attached an Analytica model, which you can download, alter with your own assumptions.  The model does not account for the second-tier prizes, which would probably have a sizeable impact on the ROI calculation.  To test yourself, you can expand the model and add in those factors.  (Clue: Use an index, Prize_type, and add this to both odds_of_winning and Payout).  And please do not construe any of this as an investment recommendation!

Expected ROI vs Mean ROI

The term "expected" is often used in mathematical communities to signify the statistical mean, so that a statistician would often refer to the graph above as showing the expected ROI.  The lottery example provides an illustrative example of this unfortunate terminology.  If you do buy a ticket for this lottery, your "expected" outcome in the common English-language usage of the term would be that you will lose.   In fact, you are about 200 times more likely to be struck by lightning than you are to win.  So to say that you have a positive expected return-on-investment would seem to imply that you can "expect" to win something.

Because of this difference between the colloquial usage of "expected" and the statistical usage of the word, I intentionally used the synonymous term mean ROI in this posting.  You should always be cognizant of the term "expected" when presenting your uncertain results, and the potential for mis-interpretation.

View the model

If you have a copy of Analytica, you can download the Mega-Millions ROI.ana model and modify it.

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Lonnie Chrisman

Lonnie Chrisman, PhD, is Lumina's Chief Technical Officer, where he heads engineering and development of Analytica®. He has authored dozens refereed publications in the areas of machine learning, Artificial Intelligence planning, robotics, probabilistic inference, Bayesian networks, and computational biology. He was was in eighth grade when he had his first paid programming job. He was awarded the Alton B. Zerby award "Most outstanding Electrical Engineering Student in the USA", 1987. He has a PhD in Artificial Intelligence and Computer Science from Carnegie Mellon University; and a BS in Electrical Engineering from University of California at Berkeley. Lonnie used Analytica for seismic structural analysis of an extension that he built to his own home where he lives with his wife and raised four daughters: So, he really trusts Analytica calculations!