Question

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

• A. $Eand{F}^C$(the complement of the event $F$) are independent
• B. $E^{C}and{F}^C$ are independent
• C. $P\left(\frac{E}{F}\right)+P\left(\frac{E\text{'}}{F\text{'}}\right)=1$
• D. All of the above

Answer 1

The correct option is D

All of the above

The explanation for the correct option:

Step 1: Find the intersection of the 2 independent events

It is given that $EandF$ are independent:

We know that, if $EandF$ are independent,

$P(E\cap F)=P\left(E\right).P\left(F\right)$

Step 2: Find the intersection of $Eand{F}^C$

Thus,

$\begin{array}{rcl}P(E\cap F\text{'})& =& P\left(E\right)–P(E\cap F)\\ & =& P\left(E\right)(1-P(F\left)\right)\\ & =& P\left(E\right).P(F\text{'})\end{array}$`

Step 3: Find the intersection of $E^{C}and{F}^C$

Since $E^{C}and{F}^C$ are independent

$\begin{array}{rcl}P(E\text{'}\cap F\text{'})& =& 1-P\left(EUF\right)\\ & =& 1-\left[P\right(E)+P(F)-P(E\cap F\left)\right]\\ & =& [1-P(E\left)\right][1-P(F\left)\right]\\ & =& P(E\text{'})P(F\text{'})\end{array}$

Step 3: Find the correct answer

$\begin{array}{rcl}P\left(\frac{E}{F}\right)+P\left(\frac{E\text{'}}{F\text{'}}\right)& =& 1\\ P\left(\frac{E}{F}\right)& =& P\left(E\right),and\\ P\left(\frac{E\text{'}}{F\text{'}}\right)& =& P(E\text{'})\\ Thus,P\left(E\right)+P(E\text{'})& =& 1\end{array}$

Hence, option (D) is correct answer.