Question

If $A$ and $B$ are two independent events such that $P(A\cap B\text{'})=\frac{3}{25}$, and $P(A\text{'}\cap B)=\frac{8}{25}$, then $P\left(A\right)=$

• A. $\frac{1}{5}$
• B. $\frac{3}{8}$
• C. $\frac{2}{5}$
• D. $\frac{4}{5}$

Answer 1

The correct option is A

$\frac{1}{5}$

Explanation for the correct option:

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

Given that, $A$ and $B$ are independent events.

$\begin{array}{rcl}P(A\cap B\text{'})& =& P\left(A\right).P(B\text{'})\\ & =& \frac{3}{25}\end{array}$

$\Rightarrow$$P\left(A\right).\left[1-P\left(B\right)\right]=\frac{3}{25}$ ..…(1)

Similarly, $\begin{array}{rcl}P(A\text{'}\cap B)& =& P(A\text{'}).P\left(B\right)\end{array}$

$\begin{array}{rcl}& =& \frac{8}{25}\end{array}$

$\Rightarrow$$P\left(B\right).\left[1-P\left(A\right)\right]=\frac{8}{25}$ ..…(2)

Step 2. On solving equation (1) and (2), we get

$$\begin{gathered} P\left(A\right)=\frac{8}{25}-\frac{3}{25} \\ P\left(A\right)=\frac{5}{25} \\ P\left(A\right)=\frac{1}{5} \end{gathered}$$

Hence, Option ‘A’ is Correct.