In a tuition batch of two students, the probability that X will pass the exam is \( \frac{2}{5} \) and that of Y is \( \frac{3}{4} \). What is the probability that neither of X and Y will pass the exam? Assume that the outcomes of exams for X and Y are independent of each other.
Step-by-step Solution:
Let \( P(X) \) and \( P(Y) \) be the probabilities that X and Y pass, respectively. \[ P(X) = \frac{2}{5} \] \[ P(Y) = \frac{3}{4} \] The probability that X fails is: \[ P(\neg X) = 1 - \frac{2}{5} = \frac{3}{5} \] The probability that Y fails is: \[ P(\neg Y) = 1 - \frac{3}{4} = \frac{1}{4} \] Since the events are independent, the probability that neither passes is the product of their individual probabilities of failing: \[ P(\neg X \cap \neg Y) = P(\neg X) \times P(\neg Y) \] \[ P(\neg X \cap \neg Y) = \frac{3}{5} \times \frac{1}{4} = \frac{3}{20} \]