Travis Wolf, PhD,
Assistant Professor of Mathematics at the University of Mary

Jane Hewitt, MD, has a problem. Ever since she read George Eliot’s novel Daniel Deronda in high school, she’s had a fascination with playing roulette. Yet whenever she plays, she loses—just like the novel’s heroine, Gwendolen Harleth. Dr. Hewitt promised her husband, Victor, she would never play roulette again, and in return he promised to completely stop smoking.

The problem is that she’s 35,000 feet in the air on a passenger jet flying to Las Vegas to present a paper at the annual meeting of the American Academy of Dermatology. Dr. Hewitt can afford to lose a few hundred dollars at the roulette table—but there’s that promise. And if she breaks it, Victor has threatened to change their last name to “Blewitt.”

The next day, Dr. Hewitt’s presentation, “New Transmission Modes for Hansen’s Disease,” goes very well. The ballroom at the Bellagio Hotel is filled with 2,000 dermatologists and related health professionals. Many colleagues congratulate Dr. Hewitt afterwards and invite her to dinner. She agrees to meet several friends who flew from India where they are treating a recent upsurge in leprosy (Hansen’s disease).

On the way back to her room, Dr. Hewitt walks through one of the Bellagio’s casinos, which can’t be avoided, and soon spots a roulette table. She slows down and magically, her shoes glide towards the spinning roulette wheel.

‘Surely,’ she thinks, ‘there’s no harm in watching for a while.’ Placing a bet is a young woman with tattoos on her left shoulder and down that arm. The inked animals catch the doctor’s interest, since tattoo needles can act as disease transmission vectors. More interesting to Dr. Hewitt is another young woman—without a dragon tattoo. She exudes the naïve excitement of a newbie gambler as she puts a $10 chip on number 7 on the roulette table.

Image of a standard roulette table.

‘Could number 7 really be lucky?’ Dr. Hewitt wonders. The roulette wheel includes numbers 1 through 36, as well as 0 and 00, in non-sequential order. Each time the wheel is spun, there is one winning and 37 losing numbers, making the odds in favor of winning 1 to 37 (or 1:37). Hence the odds against winning are 37:1.

Dr. Hewitt considers outcome probabilities for treatments in her daily medical practice, so she automatically converts these odds to a probability. With each spin of the wheel, the probability of a 7 being the winning number is determined by dividing 1 by 38, since there is only one winning number out of 38 total numbers. This equals approximately 0.026 or 2.6 percent. Dr. Hewitt’s patients would not find such a small chance of succeeding—which translates to a 97.4-percent likelihood of failure—very reassuring.

The croupier spins the roulette wheel clockwise and then rolls the ball in the opposite direction. The spinning of the wheel and ball in contrary motion mesmerizes everyone around the table—until the ball falls into the number 18 slot, which no one picked.

‘Is there a better bet?’ Dr. Hewitt asks herself. The woman without the dragon tattoo could wager that one of the red or one of the black numbers will win. After all, there are 18 black and 20 non-black numbers (18 red and two green numbers [0 and 00]). Thus the odds in favor of winning a bet on a black number are 18 to 20. This can be reduced to 9:10, which makes the probability of winning 9/19, or about 47 percent.

It seems the tattoo-less woman is thinking likewise. She places a $10 bet on the black numbers. This time she wins and beams with joy as she collects chips equaling $20.

‘If she keeps betting that way,’ Dr. Hewitt realizes, ‘at least she can stay at the table a long time.’ If she came with $1,000 and places that same bet 100 times, she can assume she will win 47 of those times, since the probability of winning on a single roll is about 47 percent. On those 47 wins, she will collect $940 in chips (since a winning bet earns double), so in the end she will only lose $60 total.

But the young woman does something extraordinary. She bets $20 on red and when she wins again, she bets the $40 on black.

“What if her streak continues?” Dr. Hewitt whispers as she draws closer to the table, her hands itching to join in. Instead, she takes out her iPhone to make several calculations. “If she wins 10 times in a row, putting all her winnings into each wager, she will earn $10,240.”

The young woman wins and puts $80 on black.

“At 14 wins, she will collect $163,840,” Dr. Hewitt says to a man who joins the onlookers, “and the 15th time will earn her $327,680.”

“What are the odds of that happening?” he asks.

“Good question,” Dr. Hewitt replies and quickly calculates. “The probability of winning 14 times in a row, with each separate bet having a 0.47 chance of success, is (0.47)14. This equals 0.000029 or about three out of 100,000. The likelihood of winning 15 times in a row would be one chance in 100,000.”

The man asks another question but Dr. Hewitt is too preoccupied with her desire to sit down at the roulette table and join the fun. “What are the odds my husband would find out?” she whispers, not loudly enough for anyone else to hear, but the question thunders through her conscience.

Dr. Hewitt looks around the casino and recognizes two dozen colleagues from the convention. It’s a big casino, she realizes, and she never met most of the medical professionals at her talk. Since all of them would recognize her, she determines it would be safe to assume that at least 100 people here would recognize her—especially as the crowd continues to grow around the roulette table.

The problem is that Victor Hewitt is a pharmaceutical salesman with a major firm and will likely visit the offices of two-thirds of the 100 dermatologists who would see her playing roulette.

She asks herself the ultimate question, ‘But would they say anything to my husband?’

People either talk or they don’t; some are chatty and others say as little as possible beyond the business at hand, Dr. Hewitt reasons. So she assumes there’s a 50-percent chance that any given doctor who speaks to her husband on his visit will mention that he or she saw her. This means the probability that the first doctor her husband sees will not say something is ½, which equals 0.5 or 50 percent. The probability that neither of the first two doctors he sees will say something is (½)2, which equals 1/4 or 25 percent.

So the odds that “what happens in Vegas stays in Vegas” when Victor visits the first doctor are even (1:1). The situation is similar to flipping a coin, where the outcome is either Heads (H) or Tails (T), with Heads as the outcome “the doctor will say something to my husband about this” and Tails as the outcome “the doctor won’t say anything about this.” Then the odds that “what happens in Vegas stays in Vegas” on the first two visits are 1:3, since if two coins are flipped, there are four possible outcomes: H-H, H-T, T-H, T-T. Only T-T equates to neither of the two doctors saying something.

So far the odds that “what happens in Vegas stays in Vegas” seem reasonable. Dr. Hewitt edges towards the roulette table, just as the young woman wins for a third time in a row and immediately bets all her winnings on red.

Image of a standard roulette wheel.

A man and a woman on the other side of the roulette table wave at Dr. Hewitt. She waves back, recognizing them from medical school. They are also Victor’s clients who, she reminds herself, will visit about 67 doctors who would take notice if she starts gambling.

Now the numbers get intimidating, she realizes. The probability that none of these 67 doctors will say something about her is (½)67. Dr. Hewitt recalls that the probability the first doctor Victor sees will not say something is ½, which equals 50 percent. It’s a coin flip. The probability that this outcome will be the same 67 times in a row is ½ x ½ x ½ … (67 times in total). The first step in making this calculation is to determine 267, which equals 147,573,952,589,676,412,928, or almost 150 quintillion.

That’s a big number—so big it’s about the same as the number of grains of sand on planet Earth. However, instead of heading to the beach with a sifter, Dr. Hewitt knows this is part of a fraction: 1/147,573,952,589,676,412,928 or 1 divided by almost 150 quintillion, which amounts to 0.000000000000000000678 percent—a really small number that also indicates how massively unlikely it is that none of the 67 doctors will mention her.

Thus, the likelihood that at least one of the 67 doctors will mention that they saw Dr. Hewitt is 99.9999999999 9999999932 percent. The odds that “what happens in Vegas stays in Vegas” are 1:147,573,952,589,676,412,927, the likelihood of which approaches an impossible event.

"The Casino, Monte Carlo," by Christian Ludwig Bokelmann (1844-1894). Bokelmann was a German genre painter in the realistic and naturalistic styles. He met an unlikely demise after falling off a ladder in his studio, while attempting to hang a laurel wreath that had been given to him by his students on the occasion of his 50th birthday. What are the odds?

“The Casino, Monte Carlo,” by Christian Ludwig Bokelmann (1844-1894). Bokelmann was a German genre painter in the realistic and naturalistic styles. He met an unlikely demise after falling off a ladder in his studio, while attempting to hang a laurel wreath that had been given to him by his students on the occasion of his 50th birthday. What are the odds?

Doing one last computation—certainly one of the most important—Dr. Hewitt reasons that if she simply doesn’t play roulette, the probability that no one will tell her husband is 100 percent.

“Too bad Gwendolen Harleth didn’t realize the odds were stacked against her before losing her
fortune in the George Eliot novel,” Dr. Hewitt says as the roulette wheel spins. A loud “Oh no” from the crowd sounds as the ball lands on black, meaning the young woman just lost all her winnings.

“What happens in Vegas stays in Vegas,” is one of the most famous taglines in advertising history, which was initiated in 2003. Las Vegas is proud of the fact that this ad campaign has succeeded in attracting millions of visitors to “Sin City.”

Mathematically, however, “what happens in Vegas stays in Vegas” is only true if nothing happens in Vegas.

Dr. Hewitt walks away from the roulette table toward the elevators to her room. She wants to call Victor before dinner to tell him how the talk went. Dr. Hewitt was nervous all week and her husband kept reassuring her that it wasn’t a coin flip. She worked so hard on her paper, he said, that success was certain.