X and Y are two independent event. If \(P(X) = \frac{1}{5}\) and \(P(X \cap Y) = \frac{2}{15}\), find \(P(Y)\).
Answer Details
The formula to find the probability of the intersection of two independent events is given by:
$$P(X \cap Y) = P(X) \times P(Y)$$
In this case, we know that:
$$P(X) = \frac{1}{5}$$
$$P(X \cap Y) = \frac{2}{15}$$
Substituting the values into the formula, we have:
$$\frac{2}{15} = \frac{1}{5} \times P(Y)$$
Solving for $P(Y)$, we get:
$$P(Y) = \frac{\frac{2}{15}}{\frac{1}{5}} = \frac{2}{15} \times \frac{5}{1} = \frac{2}{3}$$
Therefore, the probability that event Y occurs is $\frac{2}{3}$.
Answer: (a) \(\frac{2}{3}\).