Hey, Jimmy, Egg Russian Roulette is rigged…in your favor!

Hey, Jimmy, Egg Russian Roulette is rigged…in your favor!
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My wife and I were sitting around the kitchen table one night watching video clips from The Tonight Show with Jimmy Fallon on YouTube when we came across clips of a segment of the show called Egg Russian Roulette. For those who are not familiar with it, Egg Russian Roulette is a game that Jimmy Fallon plays with a guest of his show. The guest and Jimmy take turns picking eggs from a carton and smashing them on their heads. The carton contains a dozen eggs, four of which are raw and the rest are boiled. Neither Jimmy nor the guest knows which eggs are raw and which are boiled. The first person to pick two raw eggs is the loser of the game. Of course, for the audience, the issue of who will win the game is of minor import compared to the entertainment value of seeing famous people with raw yolk and albumin running down their hair and faces.

After watching celebrities like Anna Kendrick, Tom Cruise, and Bradley Cooper, in addition to Jimmy himself, smash raw eggs on their heads, my wife wondered whether the guests of the show, who get to move first in the game, are inherently at a disadvantage and more likely to lose. Given the popularity of Jimmy Fallon, I figured the answer to this question must already reside somewhere on the Web. A quick Google search took me to a reddit page on this subject, from which I learned that the guest has a 44.44% chance of winning the game (one of the comments on the page asserted that the guest’s probability of winning is precisely 4/9).

It occurred to me that the calculation of the exact probability that the guest will win the game would be an excellent applied probability problem to talk about with my mathematical economics students. Therefore, I set about deriving this probability myself. In fact, I decided to generalize this problem somewhat and ask what the guest’s probability of winning would be if the number of raw eggs in the carton initially is not four but some other number between three and eleven (the game with less than three raw eggs may not produce a winner). The results are given below (see here for details of the derivations).

As the numbers show, except for the case in which there are three raw eggs in the carton to begin with, Egg Russian Roulette favors the player who gets to go second. The likelihood that Jimmy’s guest—who makes the first selection—would be the first to smash two raw eggs on his or her head increases as the number of raw eggs in the carton rises.

Since Jimmy and his guest would have an equal chance of winning the game if the carton initially contained three raw eggs, one may wonder whether The Tonight Show should modify the game and go with three raw eggs instead (why did the show pick four raw eggs anyways?) for the sake of fairness. However, fairness may not be the most important objective for the show, since seeing celebrity guests of Jimmy’s smashing raw eggs on their heads may be what the audience want to see the most. Moreover, from a programming perspective, starting the game with three raw eggs would make it longer in that it would take more turns on average before someone loses the game. The table below shows—assuming the game is played a large number of times—the average number of turns it would take for the game to produce a loser for different numbers of raw eggs in the carton to start the game (see here for the derivations of these numbers). There is thus a tradeoff between making the game fair on the one hand and not letting the game drag on too long and thereby losing the audience on the other hand. Given the twin considerations of fairness and the length of the segment, maybe The Tonight Show made a wise decision to use four raw eggs in Egg Russian Roulette.

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