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    Question

    A doctor knows that 90% of patients have flu and 10%

    have allergy. Rash appears in 8% of flu patients and in 95% of allergic ones. If a patient shows rash, what is the probability they have an allergy?
    A 0.1523 Correct Answer Incorrect Answer
    B 0.3579 Correct Answer Incorrect Answer
    C 0.5689 Correct Answer Incorrect Answer
    D 0.9256 Correct Answer Incorrect Answer

    Solution

    Let the events be:

    • F: patient has flu → P(F) = 0.90
    • A: patient has allergy → P(A) = 0.10
    • R: patient shows rash
    Conditional probabilities:
    • P(R | F) = 0.08
    • P(R | A) = 0.95
    We are asked to find: P(A | R) = Probability that a patient has allergy given that they have rash Apply Bayes’ Theorem: P(A | R) = [P(A) × P(R | A)] / P(R) Where: P(R) = P(F) × P(R | F) + P(A) × P(R | A) = (0.90 × 0.08) + (0.10 × 0.95) = 0.072 + 0.095 = 0.167 Now plug into Bayes' formula: P(A | R) = (0.10 × 0.95) / 0.167 = 0.095 / 0.167 ≈ 0.568

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