The Blacklist: Statistics and The King Family

On a recent episode of the blacklist, the central plot device was centered around a bizarre family whose scion waged repeated battles to the death to secure their inheritance. These battles were in the form of auctions for illegal goods. Whichever brother secured for the family the fewer proceeds would be required to play one round of Russian Roulette.

The problem with this becomes immediately apparent with only a small amount of thought about the statistics involved.

It's clear from the episode that the game is the "classic" variant - the cylinder is spun each time. Clearly if it weren't, there would be a fatality within 6 rounds (although it's possible the revolver in the episode was not a six-shooter, they're by far the most common, so we'll assume here that the gun has six positions and that the referee loaded only one bullet).

But when you spin the cylinder, you randomize its position, meaning that the gun has no memory of past events at all (in truth, the gun is not designed for random selection, but we can assume that the spin is performed with enough variability to offer some imprecision, and thus, unpredictability). That means that the odds of being shot in any one round are 1 in 6 - 16.6%

Fairly unlikely? Sure. But play often enough, and those odds start catching up rapidly. To make the picture easier, let's turn it around. The odds of surviving are 5 in 6 every round. At the start, there is a 100% chance of survival if you do nothing at all. 5 out of 6 times, you pull the trigger, you survive - 83.3% of the time. Spin the cylinder and do it again, and now your odds are 83.3% of 83.3%. That works out to 70%. After four rounds, your odds are 50-50. After ten, your odds of survival are now one in six.

So for the episode to ring true, either the King Family held vastly fewer auctions than is implied, or the family started out with far more progeny to whittle through, or they were extraordinarily fortunate.

In the end, Red Reddington picks up the revolver and shoots King Sr. with it - the random selection this time coming up trumps. Red laughs at the irony, shouting "what are the odds?" Pretty good, it turns out.