Russian Egg Roulette

Sometimes probability-theoretic gems are right around the corner, and Egg Russian Roulette is such a gem. A few years ago, this game was played daily on the popular NBC Tonight Show hosted by Jimmy Fallon. Usually the guest he played it with was a celebrity from sports or film. Counted among his guests were Magic Johnson, David Beckham, Tom Cruise, and Jodie Foster, to name but a few. The game works as follows: the guest and Jimmy take turns taking an egg from the carton and crack it on top of their own respective heads. The carton contains 12 eggs, of which 8 are boiled and 4 are raw. The first person who cracks two raw eggs on their head has lost the game. It’s a silly game, though a popular one among the general audience: no better entertainment than schadenfreude about a celebrity who’s got raw egg dripping from their forehead. By convention it is the guest who has to take an egg and crack it on their head first. Does this mean that the guest is playing with a disadvantage? The answer is ‘yes’: the person who takes the first egg will lose the game with probability 5/9, which will be explained further later. The calculations also produce the mildly surprising result that when one uses 9 boiled and 3 raw eggs, the game is fair, and there is no advantage or disadvantage in being the first to take an egg. 

“Hopefully the upcoming edition in July 2021 will not be cancelled, and readers of this article will have ample time to register as a contestant and represent the Netherlands in this game, like a new Raymond van Barneveld.”

Egg Russian Roulette is not just a whimsical game from an American TV show. It has a rich history, dating back to the Middle Ages. In the rural English hamlet of Swaton (184 inhabitants, currently), the throwing of eggs started around 1322 when the new abbot of the town, who owned all of the poultry, handed out eggs to loyal churchgoers as alms. Whenever the church was cut-off from the rest of the hamlet by the sometimes overflowing local river, the eggs were chucked to the churchgoers waiting on the other side of this watercourse. Recently, this tradition has been slightly adapted and restored: every year since 2006, this little village hosts a world championship of Russian Egg Roulette, which attracts contestants from all over the world. The game is played one-on-one, with six eggs, of which one is raw. It’s a shame that Dutch TV broadcasters remain uninterested in this spectacular event. The 2020 edition of the world championship Russian Egg Roulette in Swaton has been cancelled, because of the risk of spreading corona. Hopefully the upcoming edition in July 2021 will not be cancelled, and readers of this article will have ample time to register as a contestant and represent the Netherlands in this game, like a new Raymond van Barneveld. 

“An alternative method of course consists of actually performing the probabilistic experiment in real life.”

For any teacher of an introductory course of applied probability theory, Egg Russian Roulette is a godsend of a probabilistic problem. Before dealing with the probability problem from the Jimmy Fallon show, regard the simpler problem of six eggs and two players cracking the eggs in turn, whereby the first to crack a raw one is the loser. If there’s only one raw egg amongst the six eggs, then it does not matter which player starts: the probability of getting the raw egg, at the k-th attempt, is 5!/6! = 1/6 for every k, which means that each player has probability of losing 3 × 1/6 = 1/2. If two of the six eggs are raw, then the player who starts will lose with probability 3/5, because the probability of getting the first raw egg is: 2/6 = 1/3 at the first attempt, 4/6 × 3/5 × 2/4 = 1/5 at the third attempt and 4/6 × 3/5 × 2/4 × 1/3 × 1 = 1/15 at the fifth attempt. How would one calculate the odds of winning in the case of 12 eggs with 8 boiled and 4 raw eggs? An elegant approach to this problem is using Markov chains with two absorbing states. Define (i, a, b) as the state wherein i eggs have been taken from the carton so far, of which a is the amount of raw eggs taken by the guest so far, and b the amount of raw eggs taken by the host up until that point, whereby 0 ≤ i ≤ 11 en a + b ≤ 3. The initial state of the Markov chain is (0,0,0). The states (i, 2, 0), (i, 0, 2), (i, 2, 1) en (i, 1, 2) are absorbing, which means the process stops when it reaches any of these states. The one-step transition probabilities from the non-absorbing states are simple. From a non-absorbing state (i, a, b) with even i, the process goes to state (i + 1, a + 1, b) with probability (4 – a – b) / (12 – i), or to state (i + 1, a + 1, b) with probability 1 – (4 – a – b)/(12 – i), while for a non-absorbing state (i, a, b) with i uneven, the process goes to (i + 1, a, b + 1) with probability (4 – a – b) / (12 – i) or to (i + 1, a, b) with probability 1 – (4 – a – b) / (12 – i). The guest starts the game and the probability of losing is the sum of the probabilities of ending in one of the states (i, 2, 0) or (i, 2, 1) with i = 3, 5, 7, 9 or 11 and thus being absorbed. These absorption probabilities are found by multiplying the matrix of one-step transition probabilities with itself 11 times and then evaluating the probabilities associated with the row to which the initial state (0, 0, 0) belongs. This is how one finds the 5/9 probability that the guest will lose the game. In the same way, one can find the probability of losing 1/2 and 223/396 for three and five raw eggs in a carton of 12 eggs, respectively. Another method is using simulation (a Python script would only be a couple of lines of code).

An alternative method of course consists of actually performing the probabilistic experiment in real life. There’s a plethora of videos online of the many episodes of Egg Russian Roulette in The Tonight Show by Jimmy Fallon, with Higgins as unsurpassed sidekick, reminding one of the character Igor from the parody movie Young Frankenstein (played by Marty Feldman), with his characteristically shrill voice.

Click here to watch A fragment from The Tonight Show, where Jimmy Fallon plays Russian Egg Roulette with David Beckham.