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# What Guinness Stout says about how well you play blackjack?

13 February 2012

Three pals enjoy playing blackjack together, separately. Don thinks “the book” is blather so he buys insurance, stands on 12 versus two- or three-up, hits pairs of eights against 10s, and ... well, you know; the folks who rate patrons’ action figure he gives them 0.8 percent edge. Ben adheres rigorously to Basic Strategy, holding the house to 0.4 percent edge. Cal counts cards and estimates he gets roughly 1.25 percent advantage. Don starts at \$25 per hand and presses to \$100 or drops back depending on how the cards seem to be flowing, Ben holds to a flat \$50. Cal starts at \$25 and presses to \$250 depending on the count. They all average about \$50 per round. Their betting patterns give them different characteristic bankroll jumps per coup (measured by the math mavens as “standard deviations”). These are \$66.25 for Don, \$56.50 for Ben, and \$69.89 for Cal.

The upshots of particular casino visits exhibit no obvious correlations between how each of them fares and their modes of play. This, because volatility swamps the influence of edge for statistically few coups. To picture the effect, assume each plays 400 rounds. Combined, edge and volatility give Don 50 percent chance of being between \$1,055 in the hole and \$735 over the top, Ben 50 percent prospect of quitting from \$840 behind to \$680 ahead, and Cal 50 percent probability of finishing in the range from a \$690 deficit to a \$1,190 gain. The spans don’t coincide, but there’s a large overlap and any of the trio could succeed or fail.

Enquiring minds want to know if a statistical method is available to indicate whether performance distinctions exist among the strategies – even if they aren’t evident during particular visits. There is. A mathematical procedure called the “Student t test” can do the job, giving the chance that outcomes differ significantly or are merely the result of normal fluctuations.

The t test was devised to compare limited quantities of actual data, for instance – in casino gambling – final outcomes of 10 or 20 sessions by each of the three solid citizens. For such cases, each person’s set of wins and losses would be tabulated and the amounts used to calculate the mean profit or setback and the standard deviation of results among sessions. A t test would then be run on the figures to determine the probability that the values differ significantly.

The approach can be extended, admittedly somewhat loosely, to contrast the impact of alternate strategies using theoretical values of mean and standard deviation. For a series of blackjack hands, the mean would be a dollar figure found as the product of edge times gross wager. The standard deviation would be a value calculated based on the distribution of bet sizes and the anticipated frequency of decisions involving pushes and wins or losses of various magnitudes.

Application of the t test can be illustrated using the edge and standard deviation for each player. After a 400-round session, Don would have an expected loss of \$160 with a standard deviation of \$1,325, Ben a theoretical loss of \$80 with a standard deviation of \$1,130, and Cal an expected gain of \$250 with a standard deviation of \$1,398. A t test using these figures gives the probability of Don’s and Ben’s strategies leading to substantial differences in the 400 rounds as 64.2 percent; this is low enough to question whether, on a session basis, it’s worthwhile for Don to learn and follow Basic Strategy. Conversely, t tests yield over 99.9 percent probability that Cal’s outcomes differ significantly from either Don’s or Ben’s – validating the benefits of card counting. When the expected values and standard deviations for 10,000 rather than 400 rounds – 25 sessions – are run through a t test, the probability that Ben’s fortunes differ significantly from Don’s rises above 99.9 percent, showing that Basic Strategy is superior to intuition over prolonged play.

OK. By now, you’re apt to be wondering what Guinness stout has to do with all of this? Besides whatever you may think about a glass of it adding to or detracting from your casino experience, of course.

The t test was developed around 1900 by a chemist, William Gosset, who worked for the Guinness brewery. Claude Guinness wanted to compare batches of stout, which always varied somewhat from one to the next. His goal was to determine whether the differences were crucial consequences of factors his staff should try to identify and control, or unimportant results of natural phenomena such as inconsistencies in the properties of ingredients about which they couldn’t do anything. Gosset’s innovation was a way to answer this question using relatively limited amounts of data from small numbers of batches. Gosset published the methodology in 1908 in a scientific journal. To keep other brewers from learning that Guinness employed scientists and mathematicians to help run the plant, Gosset wrote under the pen name, “Student.” His analytical methodology quickly become a widely-used, standard element of the statistics toolkit. As such it lends practicality to this admonition by the punters’ poet, Sumner A Ingmark:
Use the data, don’t just guess,
Knowledge saves you great distress,