Class 12 Probability · Long Answer3 Marksalternating_trials
Class 12 Maths · Probability · Long Answer #103
A,B alternately roll a pair of fair independent dice until one obtains sum 10 and wins. A starts. Find both winning probabilities.
✅ Step-by-Step Solution
Final Answer: P(A)=12/23; P(B)=11/23
- Each turn succeeds with p=3/36=1/12 and fails with q=11/12.
- A wins on an odd turn: P(A)=p(1+q²+q⁴+…)=p/(1−q²).
- This is (1/12)/(1−121/144)=12/23. Since the game terminates almost surely, P(B)=1−12/23=11/23.
- The winning probabilities are 12/23 for A and 11/23 for B.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Each turn succeeds with p=3/36=1/12 and fails with q=11/12.
- [1 mark] A wins on an odd turn: P(A)=p(1+q²+q⁴+…)=p/(1−q²).
- [1 mark] This is (1/12)/(1−121/144)=12/23. Since the game terminates almost surely, P(B)=1−12/23=11/23.
- [0.5 marks] The winning probabilities are 12/23 for A and 11/23 for B.
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