Question

A box contains 8 red and N green balls. Two balls are drawn at random from it. If X is the random variable representing the number of green balls drawn and \(E(X) = 1.2\), then \(N = \ldots\)

• A. \(4\)
• B. \(8\)
• C. \(12\)
• D. \(16\)

Hint:

\(E(X)\) for drawing 2 balls is \(2\times\frac{N}{N+8}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
\(X\) counts the green balls in two draws. For each draw the chance of green is \(\dfrac{N}{N+8}\) by symmetry, so the expectation adds up.

Step 2: Key Formula or Approach
\[ E(X)=2\cdot\frac{N}{N+8} \]

Step 3: Detailed Explanation
Set \(\dfrac{2N}{N+8}=1.2\): \(2N=1.2N+9.6\), so \(0.8N=9.6\) and \(N=12\).
Check with the distribution: \(P(X=1)=\dfrac{8\cdot12}{\binom{20}{2}}=\dfrac{96}{190}\), \(P(X=2)=\dfrac{66}{190}\), and \(E(X)=\dfrac{96+132}{190}=1.2\).

Final Answer:
\(N=12\), option (C). \[ \boxed{12\ \text{(C)}} \]