$E_1$ , $E_2$ , $E_3$ are mutually exclusive events, each with positive probability, that together make up the sample space, and $A$ is any event. Read the two statements and choose the correct option.
I. $P(A)$ cannot be larger than the largest of $P(A \mid E_1)$ , $P(A \mid E_2)$ and $P(A \mid E_3)$ .
II. The theorem of total probability holds only when $P(E_1) = P(E_2) = P(E_3)$ .

Only ICorrect answerYour answerOnly IICorrect answerYour answerBoth I and IICorrect answerYour answerNeither I nor IICorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

I: $P(A) = \sum P(E_i)P(A \mid E_i)$ is a weighted average of the $P(A \mid E_i)$ , with weights adding to 1, so it lies between the smallest and the largest of them. True.
II: the weights $P(E_i)$ can be any positive numbers adding to 1. False.

Worth noting

The theorem needs a partition, not equal weights; unequal weights are exactly what it handles.

Ten questions from Conditional Probability & Bayes' Theorem, at your own pace.

Explanation