Q258 / 270 · NDA 2025 (Phase 2) · Mathematics
The probability that a student passes Physics test is $2/3$ and the probability that he passes both Physics test and English test is $11/15$. The probability that he passes at least one test is $4/5$. What is the probability that he passes English test?
- A$11/15$
- Correct answerB$13/15$
- C$14/15$
- D$1$
Solution
Let $P$ be the event of passing the Physics test and $E$ be the event of passing the English test.
Given:
$P(P) = \dfrac{2}{3}$
$P(P \cap E) = \dfrac{11}{15}$
$P(P \cup E) = \dfrac{4}{5}$
Using the addition theorem of probability:
$P(P \cup E) = P(P) + P(E) - P(P \cap E)$
Substituting the given values:
$\dfrac{4}{5} = \dfrac{2}{3} + P(E) - \dfrac{11}{15}$
$P(E) = \dfrac{4}{5} - \dfrac{2}{3} + \dfrac{11}{15}$
Taking the LCM of the denominators, which is $15$:
$P(E) = \dfrac{12 - 10 + 11}{15}$
$P(E) = \dfrac{13}{15}$
Answer: $13/15$
Given:
$P(P) = \dfrac{2}{3}$
$P(P \cap E) = \dfrac{11}{15}$
$P(P \cup E) = \dfrac{4}{5}$
Using the addition theorem of probability:
$P(P \cup E) = P(P) + P(E) - P(P \cap E)$
Substituting the given values:
$\dfrac{4}{5} = \dfrac{2}{3} + P(E) - \dfrac{11}{15}$
$P(E) = \dfrac{4}{5} - \dfrac{2}{3} + \dfrac{11}{15}$
Taking the LCM of the denominators, which is $15$:
$P(E) = \dfrac{12 - 10 + 11}{15}$
$P(E) = \dfrac{13}{15}$
Answer: $13/15$
Hence, the correct answer is option: 'B'.