One bullet, six chambers

A single bullet is loaded into a 6-chamber revolver. The barrel is spun randomly. Two players take turns pulling the trigger (pointed at themselves) without re-spinning. The gun eventually fires and that player loses.

Revolver Cylinder

What do you think?

Should you go first or second?

This surprises most people. The first player faces a 1/6 chance in round 1, and many assume this gives them higher overall risk. But the second player faces the same cumulative risk across their rounds. Run the simulation to confirm:

Fixed Barrel — No Re-spin

Why it's 50/50

Once the barrel is spun, the bullet's position is fixed. The six chambers are pulled in order: 1, 2, 3, 4, 5, 6.

Player 1 pulls chambers 1, 3, 5
Player 2 pulls chambers 2, 4, 6

The bullet is equally likely to be in any chamber. Three chambers belong to each player: $P(\text{Player 1 loses}) = 3/6 = 1/2$ .

With a fixed barrel, the randomness happened once (when the barrel was spun). After that, the outcome is determined — we just don't know which chamber has the bullet. The order of pulling doesn't change the probability.

Variant: re-spin after every pull

Now change the rules: spin the barrel again after each trigger pull. Should you go first or second?

What do you think?

With re-spinning, should you go first or second?

Re-spin Every Round

The re-spin variant changes everything. Each round is independent (the barrel is fresh), and the first player always carries the initial 1/6 risk.

Deriving P(Player 1 loses) with re-spin

p = P(\text{Player 1 loses})

Let p be the probability that the first player loses.

Step 1 of 5

Geometric Series Connection

The first player loses in rounds 1, 3, 5, 7, ... Each "round pair" requires both players to survive the previous pair. The probability is $\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2k} = \frac{1/6}{1 - 25/36} = \frac{6}{11}$ .

The individual terms of this geometric series are shown below. Each bar represents the probability that Player 1 loses in that specific round — try varying the number of chambers:

Geometric Series Breakdown

Chambers: 6

Sum = 0.1667 / (1 − 0.6944) = 54.55%

P₁ loses on rounds 1, 3, 5, ... — each term smaller by factor (0.833)² = 0.6944

Variant: two random bullets

Now suppose 2 bullets are placed randomly (not necessarily adjacent) in the 6 chambers. Your opponent goes first and survives. You're given the option: spin the barrel or don't spin?

What do you think?

Two random bullets. Opponent survived round 1. Should you spin the barrel?

Two Random Bullets — Should You Spin?

Spin the barrel

Don't spin

Why spin is better: After your opponent survives, you know their chamber was empty. The remaining 5 chambers still contain both bullets. Without spinning, your risk is $2/5 = 0.40$ . With spinning, you get a fresh draw from all 6 chambers: $2/6 = 0.33$ . Spin!

Variant: two consecutive bullets

What if the two bullets are in adjacent chambers? Your opponent survives round 1. Now should you spin?

Consecutive Bullets Layout

Chambers 1–4 empty, 5–6 have consecutive bullets. If opponent survived, they were at 1, 2, 3, or 4.

What do you think?

Two consecutive bullets. Opponent survived. Should you spin?

Two Consecutive Bullets — Should You Spin?

Spin the barrel

Don't spin

Why 'don't spin' wins with consecutive bullets

\text{Label empty chambers 1, 2, 3, 4 and bullet chambers 5, 6}

The two bullets are in consecutive positions (5 and 6).

Step 1 of 6

With random bullet placement, survival tells you little about your next chamber. With consecutive bullets, survival tells you a lot — you're probably far from the bullet pair, so the next chamber is likely safe too.

Summary of all variants

Variant	Strategy	P(Player 1 loss)
Fixed barrel, 1 bullet	Doesn't matter	1/2
Re-spin, 1 bullet	Go second	6/11 ≈ 54.5%
2 random bullets (opponent survived)	Spin	2/6 vs 2/5
2 consecutive bullets (opponent survived)	Don't spin	1/4 vs 1/3

Competition appearances

Russian roulette variants appear in quant interviews (the re-spin variant is a classic at DE Shaw and Citadel), as exercises for conditional probability and Bayes' rule, in game theory as a decision under uncertainty, and in Markov chain analysis since the re-spin variant is a simple absorbing chain.

Re-spin variant with a 6-chamber revolver. What's P(Player 2 loses)? (decimal like 0.45)

1/4

The takeaway

What you know about previous outcomes changes your analysis of future ones. Whether the barrel is fixed or re-spun, whether bullets are random or consecutive — each detail changes the conditional probabilities. When facing a decision under uncertainty, compute the conditional probability for each option and compare.