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

To find the value of \(n\), we need to use the formula for the probability of the union of two independent events \(X\) and \(Y\): 

\(P(X \cup Y) = P(X) + P(Y) - P(X)P(Y)\)

Given: \(P(X \cup Y) = 0.8\), \(P(X) = \frac{1}{3}\), \(P(Y) = n\).

Plugging the values into the formula:

\(0.8 = \frac{1}{3} + n - \left(\frac{1}{3}\right)n\)

To simplify, multiply the equation by 3 to eliminate the fractions:

\(3 \times 0.8 = 3 \times \frac{1}{3} + 3n - n\)

This results in:

\(2.4 = 1 + 2n\)

Subtract 1 from both sides:

\(1.4 = 2n\)

Divide by 2 to solve for \(n\):

\(n = \frac{1.4}{2}\)

\(n = \frac{7}{10}\)

Thus, the correct option is \(\frac{7}{10}\).

Answer 2

For independent events \( X \) and \( Y \), the probability of at least one occurring is given by:

\[ P(X \cup Y) = P(X) + P(Y) - P(X)P(Y). \]

Substituting the given values:

\[ 0.8 = \frac{1}{3} + n - \left(\frac{1}{3}\right)n. \]

Simplifying:

\[ 0.8 = \frac{1}{3} + n - \frac{n}{3}. \]

Combining terms:

\[ 0.8 = \frac{1}{3} + \frac{2n}{3}. \]

Multiplying the entire equation by 3:

\[ 2.4 = 1 + 2n. \]

Rearranging:

\[ 2n = 1.4 \implies n = 0.7 = \frac{7}{10}. \]