Bayes' theorem for two disjoint exhaustive hypotheses E₁ and E₂ states

Question

Accepted Answer

B. {'text': 'P(E₁|A) = P(E₁)P(A|E₁) / (P(E₁)P(A|E₁) + P(E₂)P(A|E₂))', 'label': 'B'}

1. Bayes' theorem inverts conditional probabilities using the law of total probability.
2. Denominator is total probability P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂).
3. Numerator is joint P(E₁ ∩ A) = P(E₁) × P(A|E₁).
4. Dividing gives P(E₁|A) = P(E₁)P(A|E₁) / P(A).
_Source: NCERT Class 12 Mathematics, Ch 13 "Probability", §13.5_

Answer

{'text': 'P(E₁|A) = P(A|E₁) × P(E₁) always regardless of the value of P(A)', 'label': 'A'}

Answer

{'text': 'P(E₁|A) = P(E₁) × P(A) / P(A|E₁) using the classical multiplication rule', 'label': 'C'}

Answer

{'text': 'P(E₁|A) equals 1 − P(E₂|A) whenever E₁ and E₂ partition the sample space', 'label': 'D'}