Saturday, June 25, 2016

Bernie Madoff and Powerball

The real Bernie Madoff in his new abode

Last month, I watched a rerun of the ABC miniseries about Bernie Madoff.  The series showed Richard Dreyfuss playing Madoff riding high while financial data analyst, Harry Markopolos, was finding inconsistencies in 1999 in the returns that Madoff was giving his clients which led him to conclude that Madoff must be conducting a Ponzi scheme.  The Securities and Exchange Commission (SEC) conducted a half-ass investigation and he was cleared.  Later when the financial crisis of 2008 began, Madoff was forced to admit to his family that he was indeed running a Ponzi scheme and his sons finally turned him in.

Watching my dad play Powerball religiously made me wonder what the odds of winning are.  in the game there is one urn with 69 balls numbered one thru 69 and another with 26 balls numbered 1 thru 26.  If order is taken into account the odds of winning the grand prize is one in 35,064,160,560.  The calculations for this number are below. where 1/69 is the odds of the first number being selected, 1/68 is the probability of the second number being selected with the first number being removed from the pot.  The denominator for each of the five numbers selected decreases likewise until it is 1/65.  The fraction 1/26 is the probability of selecting the powerball. This number differs from the odds of winning the grand prize of 1 in 292,201,338. 

If order is not taken into account and sampling without replacement is conducted as in the first five numbers of the powerball, the hypergeometric distribution is used to calculate the probability of a given set of numbers being chosen without respect to order.  In this case the number of possible orders of the five numbers chosen from the urn of 69 is 5! (pronounced five factorial) which is 5 x 4 x 3 x 2 x 1 = 120.  If one divides the denominator of 35,064,160,560 by 120 one finds the stated odds of one in 292,201,338.

For the odds of selecting the Powerball only there is a question in the graphic at the top that says "why is it not 1 in 26?"  The answer that they give is that order does not matter.  More specifically one should divide the probability of just selecting the Powerball (1 in 26) by the probability of having 0 zero balls out of 5 selected from the urn of 69.  One can use the hypergeometric calculator to calculate this probability,

What does any of this have to do with Bernie Madoff.  This analysis does not prove that the Powerball is a deceptive Ponzi scheme.  It is however a legal scheme where the more people play the larger the jackpot gets.  In the end the big winner is the house as it always is in gambling.  Of course Madoff was really just the tip of the iceberg (as Powerball is just the tip of the Iceberg with gambling schemes) with Wall Street corruption as the clip below explains, but he became the boogey man because most of his victims were rich.  Harry Markopolos did his math right but it fell mostly on deaf ears partly because most people didn't understand it and the Wall Street mystique still had the country enthralled.  The purpose of this post is to at least partially explain it.

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