During one of their Brighton rendezvouses, after a round of oysters and a second bottle of champagne, Tom Haley asks Serena Frome the question every mathematician longs for her lover to utter:
"I want you to tell me something...something interesting, no, counterintuitive, paradoxical. You owe me a good maths story."
Frome ("rhymes with plume"), a twenty-something blonde blessed with the looks of Scarlett Johansson might be the last person one would expect capable of satisfying Haley's request. But readers of Booker-winning author Ian McEwan's latest novel Sweet Tooth, a Cold War-era romance of multiple deception (semi-autobiographical, some have argued), should get use to mistrusting first readings. Even her newest lover Haley, a short story author and scholar of Spenser, doesn't yet know her full story. When this conversation takes place Haley believes he is dating his liaison to the Cultural Foundation that is paying him to write the next great English novel. McEwan's heroine is actually an agent for MI5 who is sent to lure Haley into a cultural war against communism (a lá Encounter) as part of the project fittingly named Sweet Tooth. Though a fresh recruit, the service has decided that Frome's beauty and steady diet of two to three paperbacks a week makes her ideal for the job.
Too ideal, perhaps. Only a week on the assignment and the Service has a new employee on its payroll and Frome a new bedfellow. Now, as Haley eagerly awaits a "good maths story", Frome has to trawl through memories of stultifying days as a mathematics student at Cambridge in order to keep up her cover. Although no Will Hunting, when pressed by her lover, Frome manages to recall a gem of a puzzle--The Monty Hall Problem. It was "going the rounds among Cambridge mathematicians when I was there", she tells him, and, as many modern readers will know, has been the cause of more embarrassment, grief, and ire than any mathematical puzzle in human history.
Serena learns about Monty's puzzle some time before 1972, which playfully suggests that it originated among the English gowned. However, as Jason Rosenhouse details in his history The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brainteaser, the puzzle was most likely US-born. Early versions of the problem were presented by Joseph Bertrand in 1889, as the "Box Paradox", and Martin Gardner in 1959, as the "Three Prisoners Problem". Its most familiar variation, inspired by the TV game show Let's Make a Deal, was first described by UC Berkeley Professor Steve Selvin in a 1975 volume of The American Statistician. Below is how Rosenhouse paraphrases Selvin's version:
"In this little teaser we are asked to play the role of a game show contestant confronted with three identical doors. Behind one is a car; behind the other two are goats. The host of the show, referred to as Monty Hall, asks us to pick one of the doors. We choose a door but do not open it. Monty now opens a door different from our initial choice, careful always to open a door he knows to conceal a goat. We stipulate that if Monty has a choice of doors to open, then he chooses randomly from among his options. Monty now gives us the options of either sticking with our original choice or witching to the one other unopened door. Assuming that our goal is to maximize our chances of winning the car, what decision should we make?"
Haley is unfamiliar with the problem, which would be plausible in the 1970s. It was not until a headline story in the New York Times in the summer of 1991, almost one year after the response to Marilyn vos Savant's solution to the problem in the Parade turned volcanic, that the brainteaser became a cultural phenomenon. Like the thousands of readers who disagreed with Vos Savant's answer and incorrectly--many quite doggedly--insisted that the chance of a car being behind either one of the two remaining doors was equally probable, Haley refuses to accept Serena's explanation that the contestant improves his chances two-fold by switching. She uses a similar line of argument as Vos Savant's initially did, asking Haley to consider a million doors opened by Monty one-by-one until only his original choice and the last unopened door Monty has not selected remains (an informal proof that set another pack of offended mathematicians at Vos Savant's heels). After unsuccessfully trying to persuade Haley, the couple leave the argument unresolved, return home, make love, and fall asleep. A more amicable end to the debate than the one Vos Savant enjoyed.
Yet few who have been flummoxed by Monty's problem can easily let it go, and Haley is no exception. In the middle of the night, Haley is roused from sleep convinced that he understands the solution to the puzzle. "I get it! Serena, I understand how it works. Everything you were saying, it's so simple. It just popped into place, like, you know, that drawing of a what's-it cube." And here is where McEwan does something really interesting. Inspired by his revelation, Haley runs to his typewriter to dash off a story that attempts "to dramatize and give ethical dimension to a line of mathematics".
In what he produces, Haley re-imagines the problem as a tale of suspected cuckoldry, in which, after some development, a jealous husband finds himself before three hotel room doors (401, 402, and 403) and knows that his wife is behind one of the doors with another man. He has one chance to break down a door and catch her in the act of betrayal. But which door to choose? Haley's protagonist settles on 403 just as an Indian couple with several small children leave room 401. Convinced that he now has a two-thirds chance of catching her if he switches his initial choice: "He makes his run, he leaps, the door of 402 smashes open and there are the couple, naked in bed, just getting going". Haley titles the story "Probable Adultery".
Haley's work, as Serena well recognizes, is a disappointment logically and artistically. Misguided by rash enthusiasm, he has tried to house a betrayal plot within the shaky walls of a mathematical conundrum, missing the critical point that Monty does not choose a door at random and the larger point that, either way, no reader of fiction is likely to care. What will be of interest to readers is the way, through Frome's reaction to the story, McEwan masterfully turns Haley's creative failure into his own success. By unjustly blaming herself for Haley's mistake--a mistake minds as brilliant as Paul Erdös didn't need a gorgeous blonde to help them make--Serena adds to a pattern of self-condemnation in romantic relationships that has dictated the course of her young life. As with the heterosexual experiments of Jeremy Mott and the dirtied blouse left in Tony Canning's hamper, Serena is simply incapable of telling a man that he has had a silly idea. So, rather than confront Haley with her misgivings, she sheepishly rewrites the details of his hotel scene to make the story, if not artistically interesting, at least logically correct. In this way, the Monty Hall Problem becomes a surprising backdrop to the early romantic frustrations of a female spy and lover of novels.
McEwan's decision to give the most famous of probability problems such prominence in his novel suggests more than a passing interest. The structure of Monty's game, in fact, resurfaces later in the story, when a piece in the Guardian exposing MI5's backing of Haley forces Serena to make a critical and final decision--maintain the lie about her role or confess and risk losing everything. Like so many of the game show contestants that have stood before Monty's two unopened doors, Serena does not turn to a mathematical law or logical rule to get out of her predicament. She relies, instead, on intuition, which proves, in this instance, a less disastrous guide in love than it is probability.