Question

If \( X \sim B(7, p) \) is a binomial variate and \( P(X=3) = P(X=5) \) then \( p = \)

• A. \( \frac{5-\sqrt{10}}{3} \)
• B. \( \frac{\sqrt{10}-2}{3} \)
• C. \( \frac{5-\sqrt{15}}{2} \)
• D. \( \frac{\sqrt{15}-3}{2} \)

Hint:

Always check that the calculated probability value falls within the interval [0, 1].

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:

We are given a condition on a Binomial distribution probability mass function. We need to set up the equation using the formula and solve for the parameter p.

Step 2: Key Formula or Approach:

For Binomial distribution B(n, p):

P(X = k) = C(n, k) × pk × (1 - p)n-k

Step 3: Detailed Explanation:

Given n = 7 and P(X = 3) = P(X = 5).

C(7, 3) × p3 × (1 - p)4 = C(7, 5) × p5 × (1 - p)2

Substitute C(7, 3) = 35 and C(7, 5) = 21. Let q = 1 - p.

35p3q4 = 21p5q2

Divide both sides by 7p3q2 (assuming p ≠ 0 and q ≠ 0):

5q2 = 3p2

Substitute q = 1 - p:

5(1 - p)2 = 3p2

5(1 - 2p + p2) = 3p2

5 - 10p + 5p2 = 3p2

2p2 - 10p + 5 = 0

Solve for p using the quadratic formula:

p = [10 ± √(100 - 40)] / 4

p = [10 ± √60] / 4

p = [10 ± 2√15] / 4

p = (5 ± √15) / 2

Since p is a probability, 0 ≤ p ≤ 1.

√15 ≈ 3.87

(5 + 3.87) / 2 > 1   (Rejected)
(5 - 3.87) / 2 < 1   (Accepted)

So, p = (5 - √15) / 2.

Step 4: Final Answer:

The value of p is (5 - √15) / 2.