Probability that a student will succeed in I.I.T. Entrance test is 0.2 and that he will succeed in 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 the places, is

Options

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

Answer

To solve the problem, we need to find the probability that a student does not succeed in both the IIT and Roorkee entrance tests.

Let's denote:
- Event A: The student succeeds in the IIT entrance test.
- Event B: The student succeeds in the Roorkee entrance test.

From the question, we have:
- \( P(A) = 0.2 \) (Probability of succeeding in IIT)
- \( P(B) = 0.5 \) (Probability of succeeding in Roorkee)
- \( P(A \cap B) = 0.3 \) (Probability of succeeding in both IIT and Roorkee)

We want to find the probability that the student does not succeed in either test, which can be represented as \( P(A' \cap B') \), where \( A' \) and \( B' \) are the complements of events A and B, respectively.

Using the formula for the probability of the union of two events, we have:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Now, substituting the known values:
\[ P(A \cup B) = 0.2 + 0.5 - 0.3 \]

Calculating this gives:
\[ P(A \cup B) = 0.2 + 0.5 - 0.3 = 0.4 \]

Now, the probability that the student does not succeed in either test is:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(A' \cap B') = 1 - 0.4 = 0.6 \]

Thus, the probability that the student does not succeed at both places is \( \boxed{0.6} \).