Gerolamo Cardona: Games of Chance

You likely have never heard of Cardona, but if you wager on games of chance, and particularly parlays, you rely on his insights.  To say Cardona had a rough go at life would be to vastly understate reality.  Perhaps it 200px-Gerolamo_Cardano_(colour)was his “against all odds survival” that led him to his fascination with probability. His mother tried three times to abort her pregnancy with homemade cocktails and poison.  His three siblings all died of the plague. Cardona himself contracted the plague at a young age, but survived.  The love his life passed away 30 years before his own passing.  The couple had three children.  His favorite son was executed for killing his wife upon learning that he was not the biological father of their three children. His second son, was disowned for stealing from Cardona.  His third son, well, he bore false witness against his father and had Cardona incarcerated in exchange for the position of torturer in the inquisition.

In between his life’s tragedies, and numerous accomplishments in other fields, Cardona found a knack for gambling.  He amassed wealth by creating and applying the field of probability.  In 1564, he wrote Liber de ludo aleae (“Book on Games of Chance”), the first book which Image result for gambling in the 1500sexplained probability in the games of chance of the day, proper odds, and effective cheating methods.  Unfortunately, the book, which was still in print this century, was not published for over 100 years after it was written.  By that time, others had forged ahead in the same area of probability, published their work, and claimed the title as the creators of probability.  In his book, Cardona became the first to make systematic use of numbers less than zero, determined that we must multiple to find the probability of two or more unrelated events, and introduced the western world to binomial coefficients and binomial theorem.

One of the main contributions to the field of probability was his correct conclusion that the probability of an event is determined by the ratio of the number of outcomes of that event divided by the total number of outcomes.  This is the sample space.  For example, if a family has two children, what are the odds that both children are girls?  Given that there are two possible outcomes, and both are equally as likely, the possible out comes would be girl/girl, girl/boy, boy/girl and boy/boy.  From the four possible outcomes, only one scenario would result in both children being girls.  Thus, there would be a (1/4) or 25% chance that both children are girls.  Another way of looking at this is that we multiply the probabilities of any two unrelated events to determine the probability of both events happening.  Here, there is 50% chance that the first child is a girl and a 50% chance that the second child is a girl.  Thus, .50 time .50 equals .25.  This is the same method Sports books use to calculate parlay payouts.  A typical sports bettor, with a 50% prediction ability against the spread, would have a 25% chance of winning a two team parlay.  Thus the proper payout would be 3 to 1.  The sports book pays 2.6 to 1.

In short, Cardona gave us the odds.  Ironically, despite his amassed wealth by using probability to gamble effectively, Cardona remained abidingly superstitious throughout his life.

With the introduction of Binomials, Cardona gave us a way to calculate the possible outcomes from the odds.  For example, if we flip a “fair” coin ten times, what are the chances that we get seven heads?  The answer is 19%.  However, if we flip that same coin 100 times, the probability that we get 70 heads is .0039%.  Similarly, what are the odds that a handicapper that is 55% against the spread, wins seven of his next ten wagers?  The answer is 27%.  The odds that that same capper would hit 70 wins in his next 100 wagers is only 2%.  This is the proof of the Law of Large Numbers — as the number of trials increase, the outcomes of those trials  will converge on the underlying probability of the event.

The Law of Large Numbers is important to gamblers for two reasons.  First, it is the proof against the idea that a certain outcome is “due.”  That is to say there is no corollary law concerning small numbers.  Thus, if a roulette wheel hits 10 blacks in a row, it does not portend that a red number is more likely on the next spin.  That thinking, referred to as the Gambler’s Fallacy, is refuted by the Law of Large Numbers.  Second, in the world of sports betting, it follows that we can draw no statistically significant conclusions from a cappers’ record against the spread until they have enough results tallied.  We would need in excess of 400 results against the spread to be certain of the prediction abilities within a margin of error of 5 points.

In short, Cardona gave us the tools to understand that deviations from the expected outcome in the short term are insignificant.  In all matters it is dangerous, and here costly, to judge ability based on short-term results.