# Playing Russian roulette

Buffalo Bill and Butch Cassidy are fighting for the showgirl's favors. At one point, Butch proposes to Bill to play Russian roulette. They take a revolver of 6 bullets and leave only three bullets housed in consecutive chambers. Bill takes the revolver without thinking and shoots at his own temple but fortunately the camera did not contain any bullet so now it is Butch's turn. At this point, Butch has the opportunity to spin the drum before firing or directly firing the revolver without spinning it.

Which option gives you the best chance of survival?

#### Solution

The least risky option, if any, consists of shoot without spinning the drum. Since we know that Bill has not died, it means that he had one of the empty bedrooms. We know that there are 3 consecutive empty bedrooms so Bill could have touched the first, second or third. In the first two cases, the next bedroom is also empty, but if the third empty room had touched the next one would contain a bullet. This means that we have 2 out of 3 opportunities to save ourselves.

On the other hand, if we rotate the drum the probability of saving us will be 1 between two and since 1/2 <2/3, we should not turn the drum before firing.