Russian roulette presents various strategic decisions depending on the game’s rules. Let’s analyze four distinct scenarios to determine the optimal choice in each.

Variant 1: One Bullet, No Re-Spin

Setup: A single bullet is placed in a six-chamber revolver. The cylinder is spun once at the start and then remains fixed. Players take turns pulling the trigger.

Question: Should you go first or second?

Analysis:

  • The bullet’s position is fixed after the initial spin.
  • Each chamber has an equal probability of containing the bullet.
  • The sequence of trigger pulls corresponds to the sequence of chambers.

Conclusion:

  • Each player has an equal chance of encountering the bullet.
  • Therefore, it doesn’t matter whether you go first or second; the probability of being shot is equal for both players.

Variant 2: One Bullet, Re-Spin Each Turn

Setup: A single bullet is placed in a six-chamber revolver. The cylinder is spun before each trigger pull. Players take turns pulling the trigger.

Question: Should you go first or second?

Analysis:

  • Each trigger pull is independent due to the re-spin.
  • The probability of being shot on any given turn is \( \frac{1}{6} \).
  • However, the first player has a \( \frac{1}{6} \) chance of being shot immediately.
  • The second player’s chance of being shot is \( \frac{5}{6} \times \frac{1}{6} = \frac{5}{36} \), assuming the first player survives.

Conclusion:

  • The second player has a lower initial risk.
  • Therefore, it’s better to go second to minimize the chance of being shot.

Variant 3: Two Bullets in Random Chambers

Setup: Two bullets are placed randomly in the six chambers. After the first player survives, you must decide whether to spin the cylinder before your turn.

Question: Should you spin the cylinder?

Analysis:

  • If you spin: Each chamber has an equal chance of containing a bullet. With two bullets, the probability of being shot is \( \frac{2}{6} = \frac{1}{3} \).
  • If you don’t spin: The first player survived, so their chamber was empty. There are now five chambers left with two bullets. The probability of being shot is \( \frac{2}{5} = 0.4 \).

Conclusion:

  • Spinning reduces the probability of being shot from 0.4 to approximately 0.333.
  • Therefore, you should spin the cylinder before your turn.

Variant 4: Two Bullets in Consecutive Chambers

Setup: Two bullets are placed in adjacent chambers. After the first player survives, you must decide whether to spin the cylinder before your turn.

Question: Should you spin the cylinder?

Analysis:

  • If you spin: The probability of being shot is \( \frac{2}{6} = \frac{1}{3} \).
  • If you don’t spin: Given the bullets are adjacent and the first player survived, there’s a \( \frac{1}{4} \) chance of being shot.

Conclusion:

  • Not spinning reduces the probability of being shot from approximately 0.333 to 0.25.
  • Therefore, you should not spin the cylinder before your turn.

Summary

Variant Optimal Choice Probability of Being Shot
One Bullet, No Re-Spin Either Equal for both players
One Bullet, Re-Spin Each Turn Go Second Lower for second player
Two Bullets in Random Chambers Spin \( \frac{1}{3} \)
Two Bullets in Consecutive Chambers Don’t Spin \( \frac{1}{4} \)

Reference