Class 12 Probability · Long Answer3 Marksalternating_trials
Class 12 Maths · Probability · Long Answer #52
A,B alternately roll a fair die, starting with A. The first to roll 6 wins. Find both winning probabilities.
✅ Step-by-Step Solution
Final Answer: P(A wins)=6/11; P(B wins)=5/11
- Each turn succeeds with p=1/6 and fails with q=5/6; all rolls are independent.
- A wins on turns 1,3,5,…, giving P(A)=p[1+q²+q⁴+…]=p/(1−q²).
- Thus P(A)=(1/6)/(1−25/36)=6/11. The probability of endless failure is zero, so P(B)=1−6/11=5/11.
- The winning probabilities are 6/11 for A and 5/11 for B.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Each turn succeeds with p=1/6 and fails with q=5/6; all rolls are independent.
- [1 mark] A wins on turns 1,3,5,…, giving P(A)=p[1+q²+q⁴+…]=p/(1−q²).
- [1 mark] Thus P(A)=(1/6)/(1−25/36)=6/11. The probability of endless failure is zero, so P(B)=1−6/11=5/11.
- [0.5 marks] The winning probabilities are 6/11 for A and 5/11 for B.
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