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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #123

Two groups compete for control of a board, with winning probabilities0.6 and0.4. Conditional on winning, group1 introduces a product with probability0.7, group2 with0.3. Given that the product was introduced, find P(group2 won).

✅ Step-by-Step Solution

Final Answer: 2/9
  1. Let E be the observed event. The mutually exclusive exhaustive cases are group1, group2. Their prior probabilities are P(group1)=3/5, P(group2)=2/5.
  2. The conditional observation probabilities are P(E|group1)=7/10, P(E|group2)=3/10.
  3. By total probability, P(E)=(3/5)(7/10) + (2/5)(3/10)=27/50.
  4. The requested joint probability with E is 3/25=3/25.
  5. By Bayes’ theorem, the requested posterior is 3/25 / (27/50) = 2/9.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are group1, group2. Their prior probabilities are P(group1)=3/5, P(group2)=2/5.
  • [1 mark] The conditional observation probabilities are P(E|group1)=7/10, P(E|group2)=3/10.
  • [1 mark] By total probability, P(E)=(3/5)(7/10) + (2/5)(3/10)=27/50.
  • [1 mark] The requested joint probability with E is 3/25=3/25.
  • [1 mark] By Bayes’ theorem, the requested posterior is 3/25 / (27/50) = 2/9.

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