Question

If $A$ and $B$ are two independent events such that $P\left(B\right)=\frac{2}{7}$, $P(A\cup B\text{'})=0.8$, then $P\left(A\right)$ is equal to

• A. $0.1$
• B. $0.2$
• C. $0.3$
• D. $0.4$

Answer 1

The correct option is C

$0.3$

Explanation for the correct option:

Step 1. Find the value of $P\left(A\right)$:

As we know, $\mathit{P}\mathbf{(}\mathit{A}\mathbf{}\mathbf{\cup }\mathbf{}\mathit{B}\mathbf{\text{'}}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}\mathit{P}\mathbf{(}\mathit{B}\mathbf{\text{'}}\mathbf{)}\mathbf{}\mathbf{–}\mathbf{}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{\text{'}}\mathbf{)}$ …..(1)

Also, $\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{\text{'}}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}\mathbf{-}\mathbf{}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}$

Step 2. Substitute the value of $P\left(A\cap B\text{'}\right)$ in equation (1):

$P(A\cup B\text{'})=P\left(A\right)+P(B\text{'})–\left[P\left(A\right)-P(A\cap B)\right]$

$\Rightarrow$$P(A\cup B\text{'})=P\left(A\right)+P(B\text{'})–P\left(A\right)+P(A\cap B)$

$\Rightarrow$$P(A\cup B\text{'})=P(B\text{'})+P(A\cap B)$

$\Rightarrow$$P(A\cup B\text{'})=\left[1-P\left(B\right)\right]+P\left(A\right).P\left(B\right)$ ….(2) $\left[\mathbf{\because }\mathit{P}\mathbf{(}\mathit{B}\mathbf{\text{'}}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{}\mathbf{-}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{,}\mathbf{}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}\mathbf{.}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\right]$

Step 3. Substitute the value $P\left(B\right)$ and $P(A\cup B\text{'})$ in equation (2)

$0.8=\left(1-\frac{2}{7}\right)+P\left(A\right).\left(\frac{2}{7}\right)$

$\Rightarrow$ $0.8=\frac{5}{7}+\frac{2}{7}\times P\left(A\right)$

$\Rightarrow$$5+2P\left(A\right)=5.6$

$\Rightarrow$ $2P\left(A\right)=0.6$

$\mathbf{\therefore }\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{3}$

Hence, Option ‘C’ is Correct.