Assertion (A): In Bayes' theorem the events E₁, E₂, , Eₙ must be mutually exclusive and together cover the sample space. Reason (R): The denominator of Bayes' theorem is P(A) = Σ P(Eᵢ) P(A Eᵢ), and the theorem of total probability gives this sum only when the Eᵢ are mutually exclusive and cover the sample space.

Question 106

MediumOriginal question

Assertion (A): In Bayes' theorem the events $E_1, E_2, \ldots, E_n$ must be mutually exclusive and together cover the sample space.
Reason (R): The denominator of Bayes' theorem is $P(A) = \sum P(E_i)\,P(A \mid E_i)$ , and the theorem of total probability gives this sum only when the $E_i$ are mutually exclusive and cover the sample space.

Both A and R are true, and R is the correct explanation of A.Correct answerYour answerBoth A and R are true, but R is not the correct explanation of A.Correct answerYour answerA is true, but R is false.Correct answerYour answerA is false, but R is true.Correct answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

A is true: the $E_i$ form a partition.
R is true and explains A: the partition is what makes the denominator of Bayes' theorem the total probability of $A$ .

Concept: Why Bayes' theorem needs a partition
Typical timeTypical: 90 s
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The rule this testsBayes' theorem

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