How the Cards Fall
Emeritus Professor Tries His Hand at Analyzing the Game of Blackjack

They say that a lot can be learned about a lake by analyzing a liter of its water. Charlie H. Cooke, an Old Dominion University emeritus professor of mathematics and statistics, working along those same lines, is learning a lot about the card game of blackjack by analyzing just one hand.

His “serendipitous” experiment, as he calls it, focuses on a blackjack player choosing to stand on two 9s against the dealer’s up card of 7.

This has been a hand of some interest to blackjack experts through the years, and many published tips suggest that the player split the nines and play two hands against the dealer. Cooke’s investigation of the hand has turned up some surprising results, and his research article on the subject was published in a recent edition of the Journal of Computers & Mathematics with Applications.

“I became interested in this problem after I retired in 2007,” Cooke explained. First, he came across the book “Bringing Down the House: The Inside Story of Six M.I.T. Students Who Took Vegas for Millions” by Ben Mezrich. Then he saw the movie “21” that was based on the book.

This sent him looking for mathematical analyses of the game and he discovered the work of Edward O. Thorp, a mathematics professor who became famous for his computer-aided probability studies of casino games and the stock market. Thorp published the best-selling “Beat the Dealer: A Winning Strategy for the Game of Twenty-One,” in 1962.

Cooke also found a 1956 article, “The Optimum Strategy in Black Jack” by Roger R. Baldwin and colleagues, that appeared in the Journal of the American Statistical Association.
“I was inspired,” Cooke said. “This was 2008-09 and the stock market was so stressful that for diversion I turned to the building of probability models for this serendipitous hand of blackjack poker.

“Old dogs can still do old tricks after they retire,” he added with a laugh. He even wrote his own Fortran program to run the computer analysis that he wanted to do.

From Fluid Dynamics To Blackjack
Cooke taught probability and numerous other graduate and undergraduate courses - as well as advised five Ph.D. students - during his 38 years at ODU. He used release time over those years to work on projects at NASA Langley Research Center in Hampton, Va., and the Fluid and Physics Branch, U.S. Army Ballistics Research Laboratories, Aberdeen Proving Ground in Maryland. Much of his work was in computational fluid dynamics, which at first blush may seem far removed from a card game. But for an elite mathematician, almost all challenges can be framed, and solved, using numbers.

Blackjack - also called 21- is one of the world’s most popular casino games because its rules are not difficult to learn and players can turn odds in their favor by developing skills and strategies.

The central object of the game is to get a hand with a value as close as possible to 21 without going over. A hand that goes over 21 is a bust. But, as analysts such as Thorp, and now Cooke, are quick to point out, the overarching object is to beat the dealer, who is forced to continue to draw until he gets a hand of 17 (or 18 through 21) or else goes bust. For example, a player who stands on a 7 and 6 for a total of 13 may win if the dealer starts with 3 up, a face card (10 points) down for a total of 13, but must draw again and gets a 9 to go bust at 22. A small advantage for the player is that the dealer must stand if his cards total exactly 17.
It doesn’t take a beginner long to learn a central point affecting outcomes in blackjack - that there are a heck of a lot of 10-point value cards in a deck of 52 cards. Kings, queens, jacks and 10s of the four suits all count 10, which means that 16 cards, or nearly 31 percent of the deck, are 10-value cards.

That brings us back to Cooke’s serendipitous hand of the player’s two 9s against the dealer’s up card of 7.

It was Thorp’s opinion that the player should split, which means he separates the 9s and asks for each 9 to be hit with another card, allowing him to lay two bets against the dealer’s hand. This opinion is echoed by contemporary online experts, one of whom writes: “If the player is dealt two 9s, then his hand value is 18. If he splits (because of the large number of 10-value cards), then he has two hands that have a likelihood of reaching a value of 19.” The expert further writes that the player “may be inclined not to split, but computer simulations have
shown that it is worth splitting in this case.”

(On the other hand, experts are in agreement about never splitting some pairs, such as two 5s. If the player stands on two 5s, for a total of 10, he has a good chance of being hit with a 10-value card for a very good total of 20. If the player splits, then he has two hands that have a high probability of reaching the problematic total of 15.)

But the two 9s against the dealer’s up card of 7 seemed to Cooke to be a good candidate for serendipity, meaning it could turn out better than expected for the player who chooses to stand and not split. “I kept seeing advice to split, and I said, ‘Why? I don’t understand that.’”

In Pursuit of a Sure Thing
Cooke describes himself as a “bird in hand” kind of guy, one who prefers one sure thing to two possible birds in the bush. To him the player’s total of 18, with the dealer showing 7, and likely to have a 10-value down card, seemed a good thing, if not a sure thing. He wanted to know just how good the player’s expectation of winning should be.

He knew of the contentions of Thorp and others that computer simulations had proven splitting the 9s to be the better strategy. Cooke noted that past models for blackjack probability calculations have involved a procedure called “sampling with replacement.” In other words, for every draw from a well-shuffled deck, the probability of the card being a particular one is assumed to be 1 in 52.

But as cards are dealt in a game, the actual probability is determined by the cards left in the deck. In other words, the probability changes to 1 in 51, 1 in 50, 1 in 49, and so forth. To compensate for this, the correct probability model assumes that “sampling without replacement” is used.

“The problem with sampling without replacement is that it’s computationally and logically much more difficult,” Cooke said, “and, in general, sampling with replacement may be close enough to show a trend, which may be enough to satisfy a blackjack player.”

Also, certain circumstances make it nearly impossible to write a computer program to sort out a sampling without replacement problem - say if the dealer, trying to beat 18, draws a bunch of low-value cards to add to his 7 and ends up holding eight cards, which is possible. The complexity arises because sorting of possibilities encounters an inverted tree structure, with an explosion in computer logic as the number of draws increases.

Cooke, nevertheless, forged ahead with the sampling-without-replacement strategy and was able to probabilistically model the dealer’s results if he drew up to five cards in each contest against the player’s two 9s. He truncated the simulation at five draws for the complexity reasons noted above, but built a mathematical case for the simulation not needing to extend to six, seven or eight draws.

He also analyzed the hand using the traditional sampling-with-replacement model. Finally, as a gauge of whether his calculations were correct (computer programming was involved), he resorted to the use of the statistical law of large numbers applied to repeated trials of an experiment.

Here, more importantly, to check his theoretical analysis he conducted the experiment of obtaining consecutive dealer hands by playing through decks (with the two 9s and one 7 removed) until he had a statistically significant 241 experimental hands of the dealer with a 7 up card trying to beat a player with two 9s. The experimental results came within one-tenth of 1 percent of replicating the player’s expectation of winning from the sampling-without-replacement simulation. “I was surprised and delighted,” Cooke said.

Winning Money is Not the Object

So what were the essential results of Cooke’s research?

• Sampling without replacement is better because it was found that the simpler sampling with replacement leads to expectation errors of 7 percent or better if the dealer is forced to draw five cards or more to try to beat the player’s18.
• If this serendipitous hand were played continuously over long periods of time, the player is expected to strafe the dealer, winning, on average, 41 cents on each dollar gambled.

This means that the player who stands on the two 9s has a high expectation of winning, so high that splitting the 9s would seem to be an unnecessary risk. Cooke said that he believes it to be impossibly complex to program the logic required for calculating probabilities to investigate the expectation of winning for a player who split the 9s, assuming sampling without replacement.

The logic has a tree structure with an explosion of branches as the number of draws increases. For five dealer cards the probability sample space contains more than 113,000 sample points, and the size increases essentially by an order of magnitude for each successive dealer card drawn. The size of the sample space for, respectively, 3, 4 and 5 dealer cards is 1,392, 18,424 and 113,842. The greater the size, the more logic branches needed to sort out probabilities.

He said it doesn’t bother him that his findings may be too rarefied to help a casino blackjack player. “I’ve never gambled on the game. I’m not doing this for money, or for any other gain. An emeritus professor no longer faces the ‘publish or perish’ ultimatum.”

But he chuckles when he tells the story of the parlor game he has come up with as a result of his research. He has dealt through so many decks in his experiments (to finish the hand starting with the player’s two 9s and the dealer’s 7) that he now knows how to maximally stack a particular deck in favor of the player, such that the dealer never wins. Although, without stacking the player usually wins more amply than the dealer.

“I let my son-in-law have the dealer’s role while playing through the deck stacked in the player’s favor and he was greatly surprised that the player always came out with more money. If the next time he comes, I stack the deck for maximal dealer wins, and this time take the dealer’s role for myself, he will get another surprise.”

A few days after the interview conducted for this story, Cooke e-mailed news of a eureka moment that is dense enough to challenge the understanding of mere mortals, or even the average blackjack player. His analogy is to John von Neuman, at the time considered the most intelligent mathematician in the world, being hired to consult with the Rand Corp. think tank at a huge salary, but only being asked to report on what he thought about while shaving!

Cooke said even a lesser light can have an occasional von Neuman moment, and that perhaps he could be paid for what he thinks about on his daily two-mile walk.

“The thinking on the daily walk paid off,” he wrote in the e-mail. “I have been able to find an error bound that estimates the size of the error when the serendipitous hand is stopped after a given number of dealer draws. This justifies the stopping after five cards on the serendipitous hand, as the bound indicates a negligible error.

“In any situation where the player decides to stand and the dealer is required to draw several times to finish the hand, the same analysis applies. So a generally useful approach has been obtained because of my study.”

He said he hopes to be prepared to present these new results soon at a mathematics conference. Maybe there he can find an audience that can fully appreciate his intricate mind games.


Quest Summer 2010 • Volume 13 Issue 1