THE DEAL BREAKER If you're a guest on Howie Mandel's show, you should bring Jeffrey Rosenthal -- not your dad J. Kelly Nestruck, National Post Published: Tuesday, May 30, 2006 Remember asking your math teacher when you'd ever need to use any of your lessons in the real world? In our three-part series, we answer that question with popular culture. Today, Deal or No Deal. Tomorrow, The Da Vinci Code. - - - On a recent episode of the game show Deal or No Deal, a contestant named Josie received an offer of US$42,000 for a briefcase -- one that could have held more than four times that amount. The young woman had brought along her father for advice and, at that moment, turned to him for guidance. "You know something honey? Everything in life's a risk," he told her. "You can't make it to second base if you don't leave first." Taking dear old dad's folksy wisdom to heart, Josie plunged ahead fearlessly -- and ended up leaving the show with just US$25,000. If you should ever be so lucky as to end up on one of the 35 versions of Deal or No Deal that air around the world, it's probably best to leave the parents and their platitudes at home and bring along a mathematician. While the publicity material for Deal or No Deal, the American version of which airs on CH and NBC in Canada, claims it is about "nerves, instincts and raw intuition," the game has just as much to do with probability. Jeffrey Rosenthal, a professor in the Department of Statistics at the University of Toronto, has been watching Deal or No Deal out of professional interest. (For fun, he likes to watch Jeopardy!) Touring to promote his pop math book Struck by Lightning: The Curious World of Probabilities, he has received a number of requests from readers for tips on how to answer host Howie Mandel's question. His mathematical way of looking at the show clearly gets results: When I tested Rosenthal with situations from the show in his sixth-floor office in U of T's Sidney Smith Hall, he consistently outperformed the actual contestants. (See sidebar below.) For those who haven't seen the show, Deal or No Deal is played like this: A contestant picks a briefcase from 26 briefcases, in which amounts from a penny to US$1-million have been randomly distributed. The contestant then opens a number of the remaining 25 briefcases, revealing the amounts within. (To add a bit of sex appeal and perhaps addle the brains of the male contestants, they are held by beautiful models.) After each round of briefcase opening, an anonymous "banker" calls and makes the contestant an offer for the briefcase he selected at the beginning. Toronto-born host Howie Mandel then poses the titular question: Deal or no deal? At first glance, the odds of walking away with US$1-million seem fairly high. In fact, each contestant has a one in 26 chance of picking the million-dollar briefcase and a seven in 26 chance of picking one containing at least US$100,000. (Compare that to Lotto 6/49's one in 13,983,816 odds of winning the jackpot, or one in 57 odds of even getting a measly $10.) But if each contestant answered "No deal" to Mandel all the way through, the average payout would be US$131,477.54. To evaluate whether a deal is a good one or not, Rosenthal suggests you compare the banker's offer to the average of all the amounts that remain unopened. "If the banker's offer is more than the average, that means on average you'll get more money if you take it," says Rosenthal, who has consulted for online casinos wanting to make sure the odds are in their favour when developing new games. "If the average of the numbers is less, it means on average you'll get more money if you say, 'No deal.' " The problem with thinking in terms of averages, however, is that each contestant only gets to play once and most of the briefcases contain US$1,000 or less. "And there's one chance in 26 of winning just a penny," Rosenthal notes. Most contestants will usually take a poorer sure thing over a richer uncertainty once the banker's offers start going down. (Since the American version premiered last December, the biggest win has been US$464,000 and the smallest US$5.) When going for the best offer, it doesn't really matter what's in your briefcase -- the game is more like playing the stock market. "You buy a stock because you think next month somebody's going to be willing to pay a lot of money for it," Rosenthal says. "You don't really care what the company's really worth." So, your response to the question "Deal or no deal?" eventually comes down to what Rosenthal calls utility functions -- more of an economic than mathematical concept. In short, you have to decide how happy the Banker's current offer will make you, and if the possibility of being a bit happier is worth the risk of continuing to open briefcases. The "right" answer depends on the contestant's personality and how useful the offered money would be to him or her. Contestants who care less about the money -- or who care more about fame than cash -- are more likely to take the risk and win big (or lose big). That it is unlikely anyone will ever win the US$1-million prize hasn't hampered Deal or No Deal's popularity -- it's been a ratings hit everywhere. In fact, on my way out of Sidney Smith Hall after interviewing Rosenthal, I passed two University of Toronto students talking about the previous night's show over coffee. What are the odds of that? - The two-hour season finale of Deal or No Deal with a grand prize of $5,000,000 airs on June 5. knestruck@nationalpost.com - - - IF IT DEALS GOOD, DO IT Deal or No Deal? It's a tough question, but one that having a good knowledge of probability will help you answer. We asked Jeffrey Rosenthal, author of Struck by Lightning: The Curious World of Probabilities, how he would answer Howie Mandel in the following actual situations from recent episodes. All dollar amounts in U.S. currency. Example 1: There are four briefcases left unopened and the following amounts remain in play: $1,000,000, $500,000, $300,000, $50,000 and $1. You have just received an offer of $299,000 from the banker for yours. Deal or No Deal? Professor Rosenthal's answer: Deal. Calculating the average of the remaining amounts, it appears that the banker is low-balling you. On average, you'd win $370,000.20 if you said "No Deal" all the way to the end. But, since $299,000 is an awful lot of money for a university professor, Rosenthal would personally take the money. What the actual contestant answered: No Deal. The next briefcase she opened had the $1-million in it. The Banker decreased his offer to $144,000. She ended up walking away with a mere $25,000. Who played better? Rosenthal. Example 2: The amounts $300,000, $200,000, $25,000, $5,000 and $75 are still in play. The Banker is offering you $81,000. Deal or No Deal? Professor Rosenthal's answer: No Deal. The average win if you just walked away with you briefcase would be $106,015. He'd take the risk and keep going. What the actual contestant answered: No Deal. She opened the $5,000 briefcase next and the offer went up to $124,000. Who played better: Tie. Example 3: The following amounts are left: $1-milllion, $500,000, $1,000 and $400. The Banker's offer is $357,000. Deal or No Deal? Professor Rosenthal's answer: Deal. The average amount you would win by just taking your briefcase is $375,350. The Banker is only offering a little less than that, so he'd take the money and run. What the actual contestant's answered: No Deal. The next The contestant opened the $500,000 briefcase and the offer dropped to $215,000. Who played better: Rosenthal. Example 4: The contestant from example 3 now has the numbers $400, $1000 and $1-million left, and an offer of $215,000. Deal or No Deal? Professor Rosenthal's answer: No Deal. "It would be a bit gutsy," he admits. What the actual contestant's answered: Deal. When the remaining briefcases were opened, however, it turned out she was carrying $1,000,000. Who played better: Rosenthal. (c) National Post 2006