Question

Two events \(X\) and \(Y\) are such that \(P(X) = \frac{1}{3}\), \(P(Y) = n\), and the probability of occurrence of at least one event is 0.8. If the events are independent, then the value of \(n\) is:

• A. \(\frac{3}{10}\)
• B. \(\frac{1}{15}\)
• C. \(\frac{7}{10}\)
• D. \(\frac{11}{15}\)

Answer 1

The Correct Option is C

The probability of the union of two independent events \(X\) and \(Y\) is calculated using the formula: \(P(X \cup Y) = P(X) + P(Y) - P(X)P(Y)\).

We are given that \(P(X \cup Y) = 0.8\), \(P(X) = \frac{1}{3}\), and \(P(Y) = n\).

Substituting these values into the formula yields: \(0.8 = \frac{1}{3} + n - \left(\frac{1}{3}\right)n\).

To eliminate fractions, multiply the equation by 3: \(3 \times 0.8 = 3 \times \frac{1}{3} + 3n - n\).

This simplifies to: \(2.4 = 1 + 2n\).

Subtract 1 from both sides: \(1.4 = 2n\).

Solving for \(n\) by dividing by 2: \(n = \frac{1.4}{2}\).

Therefore, \(n = \frac{7}{10}\).

The correct option is \(\frac{7}{10}\).