Question

The probability that a student will succeed in the IIT entrance test is $0.2$ and that he will succeed in the Roorkee entrance test is $0.5$. If the probability that he will be successful at both the places is $0.3$, then the probability that he does not succeed at both places is

• A. $0.4$
• B. $0.3$
• C. $0.2$
• D. $0.6$

Answer 1

The correct option is D

$0.6$

Explanation for the correct answer:

Finding the value of $P(A\text{'}\cap B\text{'})$:

Let, $A$ be the event that a student succeeds in the IIT entrance test and $B$ be the event that a student succeeds in the Roorkee entrance test.

Given that, $P\left(A\right)=0.2,P\left(B\right)=0.5,P(A\cap B)=0.3$

Thus, the probability that a student does not succeed in both places $=P(A\text{'}\cap B\text{'})$

$\begin{array}{rcl}P(A\text{'}\cap B\text{'})& =& 1-P\left(AUB\right)\left[Formulaofintersection\right]\\ & \Rightarrow & P(A\text{'}\cap B\text{'})=1-\left[P\right(A)+P(B)-P(A\cap B\left)\right]\mathbf{}\left[Formulaofunion\right]\end{array}$

Now, substituting the values

$$\begin{gathered} P(A\text{'}\cap B\text{'})=1-[0.2+0.5-0.3] \\ \Rightarrow P(A\text{'}\cap B\text{'})=1-0.4 \\ \Rightarrow P(A\text{'}\cap B\text{'})=0.6 \end{gathered}$$

Therefore, the correct answer is option (D).