I'll Take 'Probability'
For $2,400, Alex:
Odds of Three-Way Tie
April 6, 2007; Page B1
Karpagam Rajagopal almost made game-show history on July 3, 2000. That's the day "Jeopardy" aired her second appearance, after she won $7,500 on the previous episode.
Entering the final round of wagering, the library specialist from Fallon, Nev., known to her friends as "Jeeks," was tied with her opponents at $5,200. The other two bet everything, a typical strategy on the long-running, long-studied TV quiz show. But Ms. Rajagopal feared she might face a pop-culture question in the category "1962," which was 25 years before she emigrated to the U.S. from India. She bet $5,000.
All three contestants correctly answered the question about the Cuban Missile Crisis, albeit with variations on the spelling of Khrushchev. The show had just missed its first-ever trio of co-champions -- by a margin of $200.
Ms. Rajagopal's loss left her to face endless second-guessing, starting with host Alex Trebek's "Darn, Jeeks, I was hoping you'd bet it all, too!" It persists to this day.
"My family and pretty much everyone from my town has never let me forget that I'm from Nevada and I didn't know how to wager," Ms. Rajagopal, 43, says in good humor.
What has become known as "Jeeks's Rule" -- in the event of a tie, bet all or nothing -- is commemorated on "Jeopardy" fan sites and the online Trivia Hall of Fame.
Last month, she was watching the show when Jamey Kirby, Anders Martinson and Scott Weiss each finished with $16,000 and became the show's first tri-champions. They got their winnings and a rematch, which got better-than-usual ratings. The AP covered the trio, and they appeared on CNN. The news outlets repeated a number released by the show's distributor, Sony Pictures Television: The probability of such a three-way tie was 1 in 25 million.
Has lightning really struck once, almost twice, in the show's run of about 5,300 episodes? Would we really have to wait about another 75,000 years -- at the show's current rate of 230 episodes per year -- for another three-way tie?
Not likely. David K. Levine, the Washington University economist who made the probability calculation at the request of the show, can't recall ever having watched "Jeopardy." Prof. Levine assumed that each dollar amount between zero and the contestant's total was an equally likely wager; that all three contestants entered Final Jeopardy, after the initial two quiz rounds, with roughly $5,000 (which he was told by the show's staff was a typical scenario); and that all three got the question right. Then, he figured Contestant 2 would have a 1-in-5,000 chance of wagering the correct amount to finish in a tie with Contestant 1. If that happened, Contestant 3, in turn, would have a 1-in-5,000 chance of choosing the bet that forced a three-way tie. Multiply those two probabilities and you get 1 in 25 million.
After that number was publicized, Prof. Levine heard from fans of the show who pointed out that wagers are placed strategically, not randomly, and in increments of $100, sometimes with $1 added on to prevent a tie. He told me, "To be honest, it's all questionable."
Sony Pictures also consulted Gary Lorden, a Cal Tech mathematician and consultant to the CBS show "Numb3rs." Prof. Lorden, a "Jeopardy" fan who submitted his more-sober estimate two days after Sony's deadline, says he was "stunned" by the 25 million number. He used a similar method to Prof. Levine's, but figured each contestant would be choosing from among 140 bets -- multiples of $100 in a range of $2,000 to $16,000. His result: 1 in 20,000.
That sounds more reasonable, but any effort to quantify the likelihood of the three-way tie will be confounded by unpredictable human nature. Who could have forecast Ms. Rajagopal's reaction to the category "1962," or Mr. Weiss's strategic, and selfless, decision to wager an amount that could force a three-way tie?
The 36-year-old computer scientist says he realized his two opponents, tied at $8,000, were likely to bet it all, and decided to bet $2,400 of his $13,600, rather than the more-common shut-out wager of $2,401. "I just thought, 'Wow, I have this opportunity to make this happen that has never happened before.' "
Skeptics who think Mr. Weiss is rationalizing a flawed bet should know he's a "Jeopardy" buff who knows the betting schemes, and who was less excited about winning than about the hope, since fulfilled, that he'd get a notation next to his name on the "Jeopardy" fan site J! Archive. Mr. Weiss saw another potential benefit: A three-way, nonzero tie meant they'd all get to keep their winnings. "I just gave away $32,000 to two people," he says.
MIT game theorist David McAdams adds, "For the front-runner, tying may be better than winning, since it allows him to play again against relatively weak competition."
In the rematch, however, Mr. Weiss led going into Final Jeopardy, but Mr. Kirby overtook him and went on to win $60,265. The three remain friendly and have kept in touch.
Mr. Weiss, who calls himself a "math guy," says human unpredictability complicates the probability calculation about this piece of "Jeopardy" history. He asks: "What are the odds there's someone else like me who would do that?"