Aspire Faculty ID #11985 · Topic: NIMCET 2025 · Just now
NIMCET 2025

Let E and F be two events such that P(E) > 0 and P(F) > 0. Which one of the following is NOT equivalent to the condition that $P(E) =P(E|F)$?

Solution

We are given the condition: $P(E) = P(E|F)$. 
 Since $P(E|F) = \dfrac{P(E \cap F)}{P(F)}$, the condition becomes: $P(E)P(F) = P(E \cap F)$, which is exactly the definition of independence of $E$ and $F$. 
Now check each option:
1) "E and F are independent" → This is exactly equivalent to $P(E)=P(E|F)$ (TRUE). 

 3) $P(F) = P(F|E)$ → Also true under independence (TRUE). 

 4) $E^c$ and $F$ are independent → Independence is preserved under complements (TRUE). 

 2) $2P(E^c)P(F^c) \ne P(E \cap F^c)$ → This statement has no relation to $P(E)=P(E|F)$ and does NOT follow from independence (NOT equivalent). 

Therefore, the option that is NOT equivalent is: Option 2.
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