Question

If $E$ and $F$ are independent events such that $0<P\left(E\right)<1$ and $0<P\left(F\right)<1$, then

• A. $E$ and $F$ are mutually exclusive
• B. $E$ and $F^{\prime }$ are independent
• C. $E^{\prime }$ and $F^{\prime }$ are independent
• D. $P\left(E|F\right)+P\left(E^{\prime }|F\right)=1$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $E$ and $F^{\prime }$ are independent
C $E^{\prime }$ and $F^{\prime }$ are independent
D $P\left(E|F\right)+P\left(E^{\prime }|F\right)=1$

If $E$ and $F$ are independent, $E$ and $F$' are also independent. However, we cannot say that they are mutually exclusive.
Also, $E$' and $F$' are independent.
Also, $P\left(E|F\right)+P\left(E^{\prime }|F\right)=\frac{P(E\cap F)+P(E^{\prime }\cap F)}{P\left(F\right)}=\frac{P\left(F\right)}{P\left(F\right)}=1$
Hence, (B), (C), (D) are correct.