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Class 12 Probability · Very Long Answer5 Marksalternating_trials

Class 12 Maths · Probability · Very Long Answer #125

A,B,C cyclically roll a pair of independent fair dice, starting with A. The first player to get sum9 wins. Find their respective winning probabilities.

✅ Step-by-Step Solution

Final Answer: P(A)=81/217; P(B)=72/217; P(C)=64/217
  1. A turn succeeds on4 of36 outcomes, so p=1/9; q=8/9 is failure probability.
  2. A wins with probability p(1+q³+q⁶+…)=p/(1−q³).
  3. B wins with probability qp/(1−q³); C wins with probability q²p/(1−q³).
  4. Since 1−q³=1−512/729=217/729, these probabilities are81/217,72/217,64/217.
  5. The respective probabilities are81/217,72/217,64/217; their sum is1, consistent with almost-sure termination.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] A turn succeeds on4 of36 outcomes, so p=1/9; q=8/9 is failure probability.
  • [1 mark] A wins with probability p(1+q³+q⁶+…)=p/(1−q³).
  • [1 mark] B wins with probability qp/(1−q³); C wins with probability q²p/(1−q³).
  • [1 mark] Since 1−q³=1−512/729=217/729, these probabilities are81/217,72/217,64/217.
  • [1 mark] The respective probabilities are81/217,72/217,64/217; their sum is1, consistent with almost-sure termination.

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