# how to win at Russian Roulette

Every time I teach stochastic processes, we discuss whether to play Russian Roulette (don’t!). In the off chance one absolutely has to play, we determine the best time to take a turn and whether it is best to spin the barrel.

I believe in teaching important life lessons in class along with operations research. But in this case, I seriously hope none of my students consider this example on Russian roulette an important life lesson. This example is good for exploring how we can quantify probabilities to confirm our intuition.

Just for fun, here are the relevant probabilities of death based on order and spinning strategy. (I like to mix it up with some dark humor in my classes).

First, consider the odds with not spinning the barrel.

Let Ei = the event that the ith person survives (based on order of play). The first person has a 5/6 chance of survival:

$P(E1) = 5/6$

If we condition on the first outcome, the second person also has a 5/6 chance of survival:

$P(E2) = P(E2|E1)P(E1) + P(E2|E1')P(E1') = (4/5)(5/6) + (1)(1/6) = 5/6$

If we condition on the first two outcomes, the third person has a 5/6 chance of survival:

$P(E3) = P(E3|E1,E2)P(E2|E1)P(E1) + P(E3|E1')P(E1') + P(E3|E1,E2')P(E2'|E1)P(E1) = (3/4)(4/5)(5/6) + (1)(1/6) + (1)(1/5)(5/6) = 5/6$

This makes intuitive sense. The bullet goes into one chamber where it is “preassigned” to one player of the game.

Next, consider the odds with spinning the barrel.

Let Ei = the event that the ith person survives (based on order of play). The first person has a 5/6 chance of survival:

$P(E1) = 5/6$

If we condition on the first outcome, the second person also has improved chance of survival:

$P(E2) = P(E2|E1)P(E1) + P(E2|E1')P(E1') = (5/6)(5/6) + (1)(1/6) = 31/36$

If we condition on the first two outcomes, the third person has an even more improved chance of survival:

$P(E3) = P(E3|E1,E2)P(E2|E1)P(E1) + P(E3|E1')P(E1') + P(E3|E1,E2')P(E2'|E1)P(E1) = (5/6)(5/6)(5/6) + (1)(1/6) + (1)(1/6)(5/6) = 191/216$

Continuing in this way, we can compute the odds of death based on each player’s order.
Without spinning the barrel, someone will lose. If every players spins the barrel prior to his/her turn, there is a 33.5% chance that everyone will walk away from the game. Such a small action greatly affects the outcome of the game, especially for those who are among the last to go.

Do not spin the barrel:

 Order P(die) 1 0.1667 2 0.1667 3 0.1667 4 0.1667 5 0.1667 6 0.1667 P(someone dies) 1.0

Spin the barrel:

 Order P(die) 1 0.1667 2 0.1389 3 0.1157 4 0.0965 5 0.0904 6 0.0670 P(someone dies) 0.665

We can see here that there is no guaranteed way to win at Russian roulette. However, going last after everyone spins the barrel lowers your probability of losing by 60%.

#### 7 responses to “how to win at Russian Roulette”

• @j_rosenbe

Not entirely staying on topic …

(I like to mix it up with some dark humor in my classes).

I find that including more jokes (even really bad ones) improves my teaching evaluations. But, I have no idea how dark humor would go over with the students.

• David K Smith

In a related way, you could consider the strategy for a three-way duel (truel!) between A, B and C – with A more accurate than B and B more accurate than C. They take it in turns to shoot (i.e not all at once). If C shoots first what should they do? (Thanks to Martin Gardner and others for this problem.)

• Luis Eduardo

Nice problem, altough I always tought spinning the barrel resets conditions so that conditioning does not apply.

• Laura McLay

@Luis, it’s true that we start over fresh with a revolver that has been spun each time. The difference is that the further you are in line, the greater chance that you won’t even have a turn (one of the previous players doesn’t survive). We have to take that into account, which gives us a conditional probability (that is the same for everyone when we reset the barrel conditions by spinning).

• Austin

The barrel on a revolver does not spin. The part that spins is the cylinder.

• johnny

It depends on who spins the cylinder. A bullet will off balance the cylinder, so with a little practice the one who spins it can decide if the gun fires or not fairly easily.

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