Question

The probability that a student passes a Physics test is (2/3) and the probability that he passes both Physics and English test is (14/45). The probability that he passes at least one test is (4/5). What is the probability that the student passes the English test?

Answer 1

Let's denote $P(A)$ - probability to pass Physics test;

$P(B)$ - probability to pass English test;

$P(A\cap B)$ - probability to pass both Physics and English tests;

$P(A \cup B)$ - probability to pass at least one test.

$P(A \cup B) = P(A) + P(B) - P(A \cap B) \Rightarrow P(B) = P(A \cup B) - P(A) + P(A \cap B)$.

$P(B) = \frac{4}{5} - \frac{2}{3} + \frac{14}{45} = \frac{36-30+14}{45} = \frac{20}{45}$.

Answer: $\frac{20}{45}$.