If A is any event associated with sample space and If $E_1, E_2, E_3$ are mutually exclusive and exhaustive events. Then which of the following are true?

(A) $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$

(B) $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$

(C) $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{i=1}^3 P(A|E_i)P(E_i)}$, $i = 1,2,3$

(D) $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$, $i = 1,2,3$

Choose the correct answer from the options given below:

Since $E_1, E_2, E_3$ are mutually exclusive and exhaustive events:

Mutually exclusive means the events don't overlap.

Exhaustive means the events cover the entire sample space, so $P(E_1) + P(E_2) + P(E_3) = 1$.

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Checking Statement (A): $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$

The first term multiplies $P(E_1)$ with $P(E_2|A)$, which mixes different events incorrectly.

This doesn't follow any standard probability formula.

Statement (A) is false.

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Checking Statement (B): $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$

This is the Law of Total Probability.

Event $A$ can occur through any of the pathways $E_1, E_2,$ or $E_3$.

The total probability of $A$ is the sum of probabilities of $A$ occurring through each pathway.

Statement (B) is true.

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Checking Statement (C): $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{i=1}^3 P(A|E_i)P(E_i)}$

The denominator $\sum_{i=1}^3 P(A|E_i)P(E_i)$ equals $P(A)$ by the Law of Total Probability.

This gives: $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{P(A)}$

This is Bayes' Theorem.

Statement (C) is true.

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Checking Statement (D): $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$

From conditional probability: $P(A|E_i) = \frac{P(A \cap E_i)}{P(E_i)}$

The given formula does not simplify to this or any known theorem.

Statement (D) is false.

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Statements (B) and (C) are correct.

Statement (B) is the Law of Total Probability.

Statement (C) is Bayes' Theorem.

The correct answer is the option stating that (B) and (C) only are true.