Brainteasers & Puzzles

Russian Roulette

One bullet, six chambers, a single spin, two players alternating. Choose to go first or second, and weigh whether the order matters.

solvedeasy1 min

A revolver has six chambers and one bullet. The barrel is spun once, then two players take turns pulling the trigger at their own head, with no further spin, until it fires. The one it fires on loses. If you may choose to go first or second, which do you pick, and what is your chance of losing?

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#One spin fixes everything

After the single spin the bullet sits in a fixed chamber, uniform over the six. With no re-spin, the trigger simply walks the chambers in order until it reaches the bullet, so the only randomness is which chamber holds the bullet.

#Count the parities

Going first, you pull chambers 1, 3, and 5, while your opponent pulls 2, 4, and 6. The bullet is equally likely in any chamber, and three of the six are odd and three are even, so you lose with probability 36=12\tfrac{3}{6} = \tfrac{1}{2} and so does your opponent.

123456
After one spin the pulls march through the chambers in order. Player one takes the odd chambers and player two the even, three each, so each loses with probability 1/2 and the choice does not matter.

#The verdict

It makes no difference. Either way you lose half the time. Re-spinning before every pull would tilt the game and make going second better, but a single spin leaves it perfectly even.