The Numbers Behind NUMB3RS (26 page)

Rather, a present-day statistician would prefer to rely on a more justifiable analysis, such as the one that says that if the null hypothesis is true (i.e., the disputed signature is authentic), then there are forty-three true signatures and thus 43 × 42/2 = 903 pairs of signatures, each pair with a presumably equal chance of having the greatest agreement. So, without considering how extreme thirty out of thirty is—just the fact that it shows the highest level of agreement between any of the 903 pairs of signatures—there is at most one chance out of 903 of those two particular signatures being
more alike
than any of the other pairs. Therefore, either a very unusual event has occurred—one whose probability is about one-tenth of one percent—or else the hypothesis that the disputed signature is authentic is false. That would surely be sufficient for Charlie Eppes to urge his brother to put the cuffs on Hetty Robinson!

USING MATHEMATICS IN JURY SELECTION

We suspect that few of our readers are criminals. And we certainly hope that you are not a victim of a crime. So most of the techniques described in this book will be things you merely read about—or see when you watch
NUMB3RS
. But there is a fair chance—about one in five, to be precise, if you are a U.S. citizen—that at least once in your life you will find yourself called for jury duty.

For many people, serving on a jury is the only direct experience of the legal system they experience firsthand. If this does happen to you, then there is a slight chance that part of the evidence you will have to consider is mathematical. Much more likely, however, if the case is a serious one, is that you yourself may unknowingly be the target of some mathematics: the mathematics of jury selection. This is where statisticians appointed by the prosecution or defense, or both—increasingly these days those statisticians may use commercially developed juror-profiling software systems as well—will try to determine whether you have any biases that may prompt them to have you removed from the jury.

The popular conception of a jury is a panel of twelve citizens, but in fact jury sizes vary from state to state, with federal court juries different again, from a low of six to a high of twelve. Although juries as small as three have been proposed, the general consensus seems to be that six is the absolute minimum to ensure an acceptable level of fairness.

Mathematics gets into the modern jury scene at the beginning of the selection process, as the 1968 federal Jury Selection and Service Act mandates “random selection of juror names from the voter lists.” (Although the act legally applies only to federal courts, it is generally taken to set the standard.) As we have seen elsewhere in this book, randomness is a tricky concept that requires some mathematical sophistication to handle properly.

One of the goals of the jury system is that juries constitute, as far as possible, a representative cross section of society. Therefore, it is important that the selection process—which, like any selection process, is open to abuse—does not unfairly discriminate against one or more particular sectors, such as minorities. But as with the issue of determining racial bias in policing (discussed in Chapter 2), it can be a tricky matter to identify discrimination, and cases that on the surface look like clear instances of discrimination sometimes turn out to be nothing of the kind.

In one frequently cited case that went to the Supreme Court,
Castaneda v. Partida
(1977), a Mexican-American named Rodrigo Partida was indicted and convicted for burglary with intent to rape in a southern Texas border county (Hidalgo County). He appealed this conviction on the grounds that the Texas system for impaneling grand jurors discriminated against Mexican-Americans. According to census data and court records, over an eleven-year period only 39 percent of people summoned to grand jury duty had Spanish surnames, whereas 79 percent of the general population had Spanish surnames. The Supreme Court held that this was sufficient to establish a prima facie case of discrimination.

The court made its determination based on a statistical analysis. The analysis assumed that if the jurors were drawn randomly from the general population, the number of Mexican-Americans in the sample could be modeled by a normal distribution. Since 79.1 percent of the population was Mexican-American, the expected number of Mexican-Americans among the 870 people summoned to serve as grand jurors over the eleven-year period was approximately 688. In fact, only 339 served. The standard deviation for this distribution worked out to be approximately twelve, which meant that the observed data showed a difference from the expected value of roughly twenty-nine standard deviations. Since a difference of two or three standard deviations is generally regarded as statistically significant, the figures in this case were overwhelming. The probability of such a substantial departure from the expected value, often referred to as the “p value”, occurring by chance was less than 1 in 10
¹⁴⁰
.

Another high-profile case was the 1968 district court conviction of the famous pediatrician Dr. Benjamin Spock, for advocating the destruction of draft cards during the Vietnam War. There were concerns over this conviction when it became known that the supposedly randomly selected pool of 100 people from which the jury was drawn in this case contained only nine women. According to public opinion polls at the time, antiwar sentiment was much more prevalent among women than men. Dr. Spock's defense team commissioned statistician (and professor of law) Hans Zeisel to analyze the selection of jury pools. Zeisel looked at the forty-six jury pools for trials before the seven judges in the district court in the two-and-a-half-year period before the Spock trial, and found that one judge, the one in the Spock case, consistently had far fewer women on his juror pools than any of the others. The p value for the discrepancy in this case was approximately 1 in 10
¹⁸
. As it turned out, this clear case of discrimination was not pivotal in Dr. Spock's successful appeal, which was granted on the basis of the first Amendment.

What both cases demonstrate is how the application of a thorough statistical analysis can determine discrimination in jury selection, to a degree well beyond the standard “reasonable doubt” threshold.

However, selection of a representative jury pool is only part of the story. The American legal system allows for individual jurors to be eliminated from the pool at the beginning of the trial on three grounds.

The first ground is undue hardship on the juror. Typically, this occurs when a trial is likely to last a long time, and may involve sequestration. In such a case, mothers of small children, owners of small businesses, among others, can usually claim release from jury service. This leads many observers to the not unreasonable conclusion that lengthy trials generally have juries largely made up of people with lots of time on their hands, such as retired persons or those with independent means.

The second ground for exclusion is when one of the protagonists can demonstrate to the court's satisfaction that a particular juror is incapable of being impartial in that particular trial.

The third ground is the one that may result in a potential juror being subjected to a detailed statistical and psychological profile. This is the so-called peremptory challenge, where both prosecution and defense are allowed to have a certain number of jurors dismissed without having to give any reason. Of course, when a lawyer asks for a juror to be removed, he or she always does have a reason—they suspect that this particular juror would not be sympathetic to their case. How do they find that out?

JURY PROFILING

Although the right of peremptory challenge does give both sides in a case some freedom to try to shape the jury to their advantage, it does not give them the right to discriminate against any protected group, such as minorities. In the 1986 case
Batson v. Kentucky
, the jury convicted James Batson, an African-American, of burglary and receipt of stolen goods. In that case, the prosecutor used his peremptory challenges to remove all four African-Americans, leaving the case with an all-white jury. The case ended up in the Supreme Court, which, based on the composition of the jury, reversed the conviction. By then, Batson was serving a twenty-year sentence. Rather than risk a retrial, he pled guilty to burglary and received a five-year prison sentence.

As always, the challenge is to establish discrimination, as opposed to the effects of chance fluctuations. In another case,
United States v. Jordan,
the government peremptorily struck three of seven African-American jurors compared with three of twenty-one whites. That meant that an African-American in the jury pool was three times more likely to be excluded as a white. The p value in this case worked out to be 0.14; in other words, such a jury profile would occur by chance roughly one in every seven occasions. The court of appeal ruled that there was insufficient evidence of discrimination.

It turns out, however, that even when illegal discrimination is ruled out, prosecutors and defenders have considerable scope to try to shape a jury to their advantage. The trick is to determine in advance what characteristics give reliable indications of the way a particular juror may vote. How do you determine those characteristics? By conducting a survey and using statistics to analyze the results.

The idea was first tested in the early 1970s by sociologists enlisted in the defense of the so-called “Harrisburg Seven,” antiwar activists who were on trial for an alleged conspiracy to destroy Selective Service System records and kidnap Secretary of State Henry Kissinger. The defense based its jury selection on locally collected survey data, systematically striking the Harrisburg citizens least likely to sympathize with dissidents. Far from the “hanging jury” that many observers expected from this highly conservative Pennsylvania city, the jury deadlocked on the serious charges and convicted the activists of only one minor offense.

Using Math to Beat the System

DOUBLE DOWN

The dealer at the blackjack table is good at her job. She jokes with the players as she deals the hands, knowing that this will encourage them to continue placing larger and larger bets. A young man sporting a goatee, long hair, and a black leather jacket comes to the table and takes an empty seat. He converts five thousand dollars into chips, and places an enormous bet on the next hand. The dealer and the other players are taken aback by the size of his bet, but the young man breaks the tension by making some remarks about his family in Moscow. He wins the hand, making a huge profit, but then, instead of playing another hand, he scoops up his chips and leaves the table. Looking for his car in the casino parking lot, he seems anxious, even afraid. Moments later he is shot and killed by an unseen assailant.

This was the opening sequence in the second-season
NUMB3RS
episode “Double Down,” broadcast on January 13, 2006. As is often the case with
NUMB3RS
, the story isn't just about the crime itself, it's about the special worlds inhabited by the victims and the suspects—in this case, the world of professional blackjack players who challenge the gambling casinos. As “Double Down” unfolded, viewers learned that the victim, Yuri Chernov, was a brilliant mathematics student at Huntington Tech, making the case a natural one for Charlie to help with. To do that, he has to delve into the workings of the real-life battle of wits—and sometimes more—that has been going in the casino blackjack world for more than forty-five years.

On one side of this battle are secretive, stealthy “card counters,” often working in teams, who apply sophisticated mathematics and highly developed skills in their efforts to extract large winnings from casinos. On the other side are the casinos, who regard card counters as cheaters, and who maintain files of photographs of known counters. The casino bosses instruct their dealers and other employees to be always on the lookout for new faces in the ranks of players who can walk away with tens of thousands of dollars in winnings in a matter of hours.

In most states,
*
players who count cards while playing blackjack are not criminals in the literal sense. But the casinos view them as criminal adversaries—cheaters, no different from the players who manipulate the chips or conspire with crooked dealers to steal a casino's money. And because of the risk of being recognized and barred from play, card counters have to act like criminals, using disguises, putting on elaborate performances to fool dealers about their true capabilities, or sneaking around, trying desperately not to be noticed.

The root cause of the casinos' difficulty is that when it comes to blackjack, an astute and suitably knowledgeable player, unlike other casino gamblers, can actually have an edge over the casino. Casinos make a profit—a generous one—by knowing the exact probabilities of winning in each game they offer, and setting the odds so that they have a slight advantage over the players, typically around 2 to 3 percent. This guarantees that, although one or two players will make a killing every now and then, the vast majority of players will not, and on a weekly or monthly basis, the casino will earn a steady profit.

In the game of craps, for example, short of actual criminal acts of cheating (manipulating chips, using loaded dice, and the like), no player can win in the long run. When honest players win, they are simply fore-stalling the losses they will eventually rack up if they come back…and back…and back. The mathematics guarantees that this will happen.

But blackjack is different. Under certain circumstances, the players have an edge. A player who can recognize when this is the case and knows how to take advantage of it can, if allowed to continue playing, capitalize on that percentage advantage to win big money. The longer the counters are allowed to play, the more they can be expected to win.

THE PROBLEM WITH BLACKJACK

In casino blackjack, each player at the table plays individually against the dealer. Both player and dealer start with a two-card hand and take turns at having the option to draw additional cards (“hit” their hand) one at a time. The aim is to get as high a total as possible (with face cards counting as 10, aces as 1 or 11), without “going bust,” that is, exceeding 21. If the player ends up with a total higher than that of the dealer, the player wins; if the dealer has the higher total, the player loses. For most plays, the payoff is even, so the player either loses the initial stake or doubles it.

The twist that turned out to be a major headache for casinos is that, in the version of the game they offer, the dealer must play according to a rigid strategy. If the dealer's hand shows a total of 17 or more, he or she must “stand” (they are not permitted to take another card); otherwise the dealer is free to hit or stand.
*
That operational rule opens a small crack in the otherwise impregnable mathematical wall that protects the casinos from losing money.

The possibility of taking advantage of the potentially favorable rules of casino blackjack was known and exploited by only a few people until the publication in 1962 of the book
Beat the Dealer
, written by Edward Thorp, a young mathematics professor. In some ways not unlike Charlie Eppes—though without an older brother asking him to help the FBI solve crimes—Thorp was beginning his career as a research mathematician, moving from UCLA to MIT (and later to the University of California at Irvine), when he read a short article about blackjack in a mathematics journal and developed an interest in the intriguing difference between blackjack and other casino games:

It turns out that there are several features of the game of blackjack that are asymmetrical in their effect on the player and the dealer, not just the dealer's “17-rule.” The player gets to see the dealer's first card (the so-called “up card”) and can take that information into account in deciding whether to hit or “stand”—that is, the player can use a variable strategy against the dealer's fixed strategy. There are other differences, too. One asymmetry very definitely in the casino's favor is that, if both the player and the dealer bust, the dealer wins. But there are asymmetries that favor the player. For instance, the player is given the opportunity to make special plays called “doubling down” and “splitting pairs,” which are sometimes advantageous. And, particularly juicy, the player gets a bonus in the form of a 3:2 payoff (rather than just “even money”) when his initial two-card hand is a “natural”—an ace and a ten (picture card or “10”)—unless the dealer has a natural, too.

Players can capitalize on these asymmetries because, in blackjack, after each hand is played, those cards are discarded. That means that, as the plays progress, the distribution of ten-value cards in the deck can change—something an astute player can take advantage of.

When Thorp published his revolutionary discoveries in 1962, the net effect of these asymmetries and other fine points was that the version of blackjack being played on the Las Vegas “strip” was essentially an even game, with very close to a zero advantage for the casino.

In an industry where the casinos had been used to having a guaranteed edge, Thorp's discovery was completely unexpected and impressive enough to make news, and it led to hordes of gamblers flocking to the blackjack tables to play Thorp's recommended strategy, which required the player to memorize certain rules for when to hit, when to stand, and so on, depending on the dealer's up-card. All of these rules were based upon solid mathematics—probability calculations that, for example, analyzed whether a player should hit when his hand totals 16 and the dealer shows an ace. Calculating the probabilities of the various totals that the dealer might end up with and the probabilities of the totals the player would get by hitting, Thorp simply compared the probability of winning both ways—hitting and standing—and instructed players to take the better of the two options—in this case, hitting the 16.

The casinos were pleased to see the increased level of business, and they quickly realized that most of these newly minted blackjack enthusiasts only played Thorp's strategy in their dreams. Many a would-be winner had difficulty remembering the finer points of strategy well enough to execute them at the right time, or even showed a lack of dedication to the mathematically derived best strategy when subjected to the harsh realities of the luck of the draw. A run of good or bad hands—perhaps losing several times in a row by following one of the basic strategy's instructions—would often persuade players to disregard Thorp's meticulously calculated imperatives.

Nevertheless,
Beat the Dealer
was a stunning success. It sold more than 700,000 copies and made
The New York Times'
bestseller list. The game of blackjack would never be the same again.

CARD COUNTING: A MATHEMATICIAN'S SECRET WEAPON

Thorp's basic strategy, the one he developed first, simply turned a profit-maker for casinos into a fair game. How did blackjack become a potential loser for the casinos and a profit-maker for mathematicians and their avid students? Thorp carefully analyzed blackjack strategy further, using some of the most powerful computers available in the early 1960s, and he exploited two simple ideas.

One idea is for the player to vary his strategy even more (when to hit or stand, whether to double down, etc.) according to the proportion of tens left in the deck. When the chances of busting are higher than usual—say, when lots of 10s and picture cards (both counted as 10) remain in the deck and the player has a poor hand, like 16 against a dealer's 10—he can intelligently revise the basic strategy by standing instead of hitting. (If there are a lot of 10-value cards left in the deck, the chances are higher that hitting on 16 will lead to a bust.) On the other hand, when the chances of busting are lower than usual—when there are relatively more low cards in the deck—players can hit in situations where they would normally stand according to the basic strategy. These changes shift the percentage advantage from zero to a small advantage for the player.

The other idea is for the player to vary the amount bet on successive hands according to the same information—the proportion of 10-valued cards remaining in the deck. Why do that? Because the proportion of 10-value cards affects the player's prospects on the next hand. For example, if there are lots of “tens” remaining in the deck, then the chances of getting a natural go up. Of course, the dealer's chance of getting a natural goes up too, but the player gets a payout bonus for a natural and the dealer doesn't. Therefore, more frequent naturals for the player and the dealer mean a net advantage for the player!

Things would have been bad enough for the casinos if Thorp had simply explained the mathematics to the readers of his bestseller. That would have put them at the mercy of players with enough mathematical ability to understand his analysis. But Thorp did more than that. He showed them how they could count cards—that is, keep a running count of tens versus non-tens as the deck is played out—to give them a useful indicator of whether the next hand would be more favorable than average or less favorable, and to what extent.

As a result, thousands of readers of Thorp's book used its instructions to become card counters using his “Tens Strategy,” and copies of the book began to appear in the hands of passengers on trains, planes, and buses arriving in Las Vegas and other parts of Nevada, where large amounts of money could be won by applying the fruits of Thorp's mathematical analysis.

The casinos were in trouble, and they immediately changed the rules of blackjack, removing certain features of the game that contributed to the player's potential to win. They also introduced the use of multiple decks shuffled together—often four, six, or even eight decks—and dealt the cards out of a “shoe,” a wooden or plastic box designed to hold the shuffled cards and show the back of the next card before it is pulled out by the dealer.

Called the “perfesser stopper” in homage to Professor Thorp, whose personal winning exploits, while not huge, were sufficient to add to the enormous appeal of his book, the multideck shoes had two effects. They enabled the casinos to shuffle the cards less frequently, so that without slowing down the game (bad for profits) they could make sure to reshuffle when a substantial number of cards were still left in the shoe. This kept the card counters from exploiting the most advantageous situations, which tend to occur when there are relatively few cards left to be dealt. Moreover, the multiple-deck game automatically shifted the basic player versus house percentage about one-half of one percent in the house's favor (mainly due to the asymmetries mentioned above). Even better for the casino, dealing from multiple decks shuffled together meant that it would typically take much longer for the Thorp counting procedure to detect an advantageous deck, and the longer it took the more likely a player was to make a mistake in the count.

There was a predictable outcry from regular blackjack players about the rules changes—but only about reducing the opportunities to make plays like “doubling down” and “splitting pairs.” So the casinos relented and reinstituted what essentially were the previous rules. But they kept the shoes, albeit with a few blackjack tables still offering single-deck games.