## Examining a Gambler’s Claims: Probabilistic Fact-Checking and Don Johnson’s Extraordinary Blackjack Winning Streak

Most gamblers don’t err on the low side when they answer questions about how much they won the night before. In this note, we examine claims about the exploits of a gambler, Don Johnson (a.k.a. the “Beast of Blackjack”), at casinos in Atlantic City as reported in the popular press. In *The Atlantic* article “The Man Who Broke Atlantic City,” Mark Bowden reported that Johnson won almost $6 million in one night at the Tropicana in Atlantic City and $5 million and $4 million over six months at the Borgata and Caesars, respectively.

I saw the title of this article when I did a recent Google search and I immediately wrote to the authors to request a copy. They were very kind to quickly write back and forward a copy. Many thanks to the authors for their speedy generosity. Unfortunately, I discovered that their article is seriously flawed in both fact and methodology.

The most significant error in this paper comes from the boundary conditions the authors assumed when they obtained Johnson’s average daily theoretical win (ADT) for one day of play. They stated, without justification, that Don Johnson would either play 720 hands or quit after having lost $500,000. These are highly non-optimal boundary conditions. Based on these boundary conditions, the authors determine an ADT of about $41,000. In fact, the optimal boundary conditions are (approximately) for Don Johnson to quit after having either won $2.4M or lost $2.6M. These quit points lead to an ADT of about $125k, with an average of about 480 hands required to hit a boundary point.

In personal communications, Johnson stated that my results above are consistent with those obtained by his mathematicians. He also stated that based on other more intangible table conditions he sometimes exceeded these quit points. Such aberrations included, for example, error-prone dealers.

Another serious problem with the author’s results comes from the distribution the authors use to model blackjack. The author’s distribution pays the values 0, 1, -1 and 1.5 based on a 1 unit initial wager, with probabilities 0.0982, 0.0483, 0.3893 and 0.4623. This distribution gives a house edge of 0.25% with a standard deviation of 0.9809. In the actual blackjack game that Johnson played, outcomes cover the entire range from winning 8 units to losing 8 units, after splits, doubles and surrender are taken into consideration. The correct house edge is 0.263% with a standard deviation of 1.1417. It follows that the authors used a distribution whose standard deviation was substantially smaller than the actual standard deviation for blackjack.

A player’s ADT playing against a loss rebate is proportional to the standard deviation of the game. Underestimating the standard deviation is another cause that led to the authors to underestimate Don Johnson’s overall win-rate. It appears the authors did not investigate the exact blackjack rules that Don Johnson played against. Had they made this effort, the authors could have gotten the correct distribution for their blackjack simulations with a simple Google search, which would have led them to the wizardofodds.com website. This search would have led them to use an accurate blackjack model.

There are other sources of positive expected value that the authors did not consider. In addition to Johnson’s ADT of $125k from a correct loss rebate strategy, Johnson was also given $50k per day in “show up money.” He also claimed to work hard to create conditions where the dealers made errors. When I heard Don Johnson speak at the World Game Protection Conference in Las Vegas in 2013, he stated that he coaxed about two or three errors per day from the dealers. That’s at least another $200k per day. These additional sources yield about an ADT of about $375k per day. Playing 40 days (as the authors assumed), it follows that Don Johnson’s overall results were completely in line with expectation. Indeed, 40x$375k = $15M.

I am surprised that the authors did not do a simple Google search on “Don Johnson blackjack.” If they had, then they would have found my work. For nearly a year, my results have been in the top 5 under the Google search “Don Johnson blackjack.” I have written a number of articles on Don Johnson as well as on the general theory of beating loss rebates. In the last few months, I have proved and published a series of theorems that I call the “Loss Rebate Theorems.” These allow a direct spread sheet solution that does not require Monte Carlo modeling.

The authors simulated 500,000″ Don Johnsons” with incorrect boundary conditions and an incorrect blackjack model. I simulated billions of “Don Johnsons” using a variety of quit points and the exact blackjack distribution corresponding to the specific game Johnson played. My simulated results were confirmed by the values produced by my Loss Rebate Theorems. I then confirmed my results were consistent with Johnson’s actual strategy by direct communication with him.

The authors state that “In the end, it is likely the case that his (Johnson’s) net winnings over the five-month period were overstated.” The truth is that Don Johnson won at a rate that was consistent with his expectation. The authors conclude that “there is a role for probabilistic fact checking at high-end periodicals.” They are right.

I invite the readers to visit my blog, http://www.apheat.net, and read my articles on loss rebates and Don Johnson.