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Amar and Baron are given a six-shot pistol that has only one bullet in its cylinder. Both of them fire a slot one after the other. A shot can be a black shot or a lethal shot. The one, who fires a lethal shot, is the winner. After each shot, the cylinder of the pistol is shuffled so that the other person has a fair chance of firing a lethal shot. It is also known that Amar had the opportunity to fire the first shot. What is the probability that Baron wins the game?

Amar and Baron are given a six-shot pistol that has only one bullet in its cylinder. Both of them fire a slot one after the other. A shot can be a black shot or a lethal shot. The one, who fires a lethal shot, is the winner. After each shot, the cylinder of the pistol is shuffled so that the other person has a fair chance of firing a lethal shot. It is also known that Amar had the opportunity to fire the first shot. What is the probability that Baron wins the game? Correct Answer 5/11

Calculation:

⇒ Let the probability that Amar wins the match be a and probability that Baron wins the match is b.

⇒ a + b = 1        ---(i)

⇒ Baron will get a chance to shoot only if Amar has a blank shot.

⇒ Now, the probability of Amar firing a blank shot is 5/6

⇒ After Amar’s blank shot, the cylinder of the pistol is shuffled. Hence, the probability that Baron shoots a lethal shot is the same as that of Amar.

⇒ b = (5/6) × a

⇒ Substituting this value in equation (i), we get,

⇒ a + (5/6)a = 1

⇒ a = 6/11

⇒ Hence, b = 5/11


Additional Information

⇒ The number of outcomes favorable to A is denoted by n(A) The total number of outcomes in sample space is denoted by n(S). Hence, the formula becomes P(A) = n(A)/n(S).

⇒ The probability of an event can vary between 0 to 1, i.e. 0 ≤ p ≤ 1.

⇒ Probability can never be negative.

⇒ Probability of occurrence of an event = 1 - (Probability that it doesn’t occur).

⇒ P(A∩B) = P(A) × P(B|A) ; if P(A) ≠ 0

⇒ P(A∩B) = P(B) × P(A|B) ; if P(B) ≠ 0

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