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 $P(E) =P(E|F)$?
🎥 Video solution / Text Solution of this question is given below:
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.