Class 12 Probability · Long Answer3 Marksalternating_trials
Class 12 Maths · Probability · Long Answer #102
A,B alternately roll a pair of independent fair dice, starting with A. A wins on sum 7; B wins on sum 10. Find P(B wins).
✅ Step-by-Step Solution
Final Answer: 5/17
- A succeeds with p=6/36=1/6; B succeeds with r=3/36=1/12. All rolls are independent.
- B wins in the first round with probability (1−p)r=5/72. A complete unsuccessful round has probability q=(1−p)(1−r)=55/72.
- Summing the geometric series gives P(B)=((1−p)r)/(1−q)=(5/72)/(17/72)=5/17.
- Thus P(B wins)=5/17; the game ends with probability 1 because q<1.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] A succeeds with p=6/36=1/6; B succeeds with r=3/36=1/12. All rolls are independent.
- [1 mark] B wins in the first round with probability (1−p)r=5/72. A complete unsuccessful round has probability q=(1−p)(1−r)=55/72.
- [1 mark] Summing the geometric series gives P(B)=((1−p)r)/(1−q)=(5/72)/(17/72)=5/17.
- [0.5 marks] Thus P(B wins)=5/17; the game ends with probability 1 because q<1.
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