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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #73

One card is lost uniformly at random from a 52-card deck. Draw three cards uniformly without replacement from the remaining deck; all are spades. Find P(the lost card was a spade).

✅ Step-by-Step Solution

Final Answer: 10/49
  1. Let E be the observed event. The mutually exclusive exhaustive cases are lost spade, lost nonspade. Their prior probabilities are P(lost spade)=1/4, P(lost nonspade)=3/4.
  2. The conditional observation probabilities are P(E|lost spade)=44/4165, P(E|lost nonspade)=286/20825.
  3. By total probability, P(E)=(1/4)(44/4165) + (3/4)(286/20825)=11/850.
  4. The requested joint probability with E is 11/4165=11/4165.
  5. By Bayes’ theorem, the requested posterior is 11/4165 / (11/850) = 10/49.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are lost spade, lost nonspade. Their prior probabilities are P(lost spade)=1/4, P(lost nonspade)=3/4.
  • [1 mark] The conditional observation probabilities are P(E|lost spade)=44/4165, P(E|lost nonspade)=286/20825.
  • [1 mark] By total probability, P(E)=(1/4)(44/4165) + (3/4)(286/20825)=11/850.
  • [1 mark] The requested joint probability with E is 11/4165=11/4165.
  • [1 mark] By Bayes’ theorem, the requested posterior is 11/4165 / (11/850) = 10/49.

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