Russian roulette game theory

Russian roulette game theory for this phone is a six-chamber cylinder with one live chamber, placed at random. Each chamber is equally likely. In a two-player duel, going first does not change your chance. The label on screen still rises after each click. There is no real gun.

The cylinder you are actually playing

When a cylinder loads, the browser picks the live chamber with cryptographic random numbers. It does not roll again on each pull.

A normal cylinder has six chambers. The first pull reads 1 in 6. A click spends an empty chamber, so the next label is 1 in 5, then 1 in 4, then 1 in 3, then 1 in 2, then the last chamber.

With seven or eight players still in, the cylinder has one chamber per player. One of them is live. A full lap cannot finish without a bang.

The table, and the rule that a bang throws the rest of the cylinder away, is on the odds page. This page uses those numbers and asks who they favour.

The chance on each pull, before anyone clicks

From the moment the cylinder loads, each of the six chambers is equally likely to be the live one. The bang is on pull 1, 2, 3, 4, 5, or 6 with chance 1 in 6 each.

The rising label is conditional. It is the chance of a bang given that the earlier pulls were clicks. After four clicks, two chambers remain, and the screen says 1 in 2. That pull was not a 1 in 2 event back when the cylinder was fresh. It was still 1 in 6, and you only reach it if the first four chambers were empty.

Should you go first?

Two players alternate. The first seat takes pulls 1, 3, and 5. The second takes 2, 4, and 6. Three chambers each. Each player has a 1 in 2 chance of being the one who gets the bang. Going first does not help or hurt.

Three players take two chambers each. The chance is 1 in 3 for every seat. Six players take one chamber each. The chance is 1 in 6 for every seat.

Four players do not split evenly. The first seat takes pulls 1 and 5. The second takes 2 and 6. The third takes pull 3. The fourth takes pull 4. The first two seats each cover 2 of 6 chambers, so 1 in 3. The last two seats cover 1 of 6. On that cylinder, going first is the worse seat.

Five players are harsher on the opener. The first seat takes pulls 1 and 6, which is 1 in 3. Each of the other four seats takes one pull, which is 1 in 6.

Seven or eight players get one chamber each. Every seat has the same chance, 1 in 7 or 1 in 8.

After a bang, the next living player starts the new cylinder. The opening seat moves with each elimination.

How to beat it

You do not beat the draw. The chamber is chosen when the cylinder loads, and the screen never shows it. Counting clicks only repeats the odds already on screen.

You do have one decision. Each player gets one Spin per game by default. A spin places the live chamber at random again and puts the label back to 1 in 6. Spend it when the label is shorter than 1 in 6. At 1 in 2, a spin moves the next pull from 1 in 2 back to 1 in 6. At 1 in 6, a spin spends itself and leaves the same label.

Fair rounds, in Settings, reloads the cylinder after every living player has pulled once. Nobody takes a second pull before the others have pulled.

That is the strategy. Use the spin when the screen is ugly. Ask for Fair rounds if the group wants one pull each before a reload. Then hold the button and let the phone decide.