Question

Let \(X\) denotes the number of times heads occur in \(n\) tosses of a fair coin. If \(P(X=4)\), \(P(X=5)\) and \(P(X=6)\) are in AP, then the value of \(n\) is

• A. 7, 14
• B. 10, 14
• C. 12, 7
• D. 14, 12

Hint:

Use binomial coefficient relationships to simplify.

Answer 1

The Correct Option is A

 To solve this problem, we need to analyze the situation where the probabilities \( P(X=4) \), \( P(X=5) \), and \( P(X=6) \) are in Arithmetic Progression (AP) for a fair coin tossed \( n \) times. Here, \( X \) represents the number of heads obtained.

1. **Probability Distribution**:

• For a fair coin, the probability of getting a head in one toss is \( \frac{1}{2} \).
• The number of heads \( X \) in \( n \) tosses follows a Binomial distribution: \(P(X=k) = \binom{n}{k} \left(\frac{1}{2}\right)^n\).

2. **Understanding AP**:

• For probabilities \( P(X=4) \), \( P(X=5) \), and \( P(X=6) \) to be in AP, we use the condition: \(2P(X=5) = P(X=4) + P(X=6)\).

3. **Forming Equations**:

• \(P(X=4) = \binom{n}{4} \left(\frac{1}{2}\right)^n\)
• \(P(X=5) = \binom{n}{5} \left(\frac{1}{2}\right)^n\)
• \(P(X=6) = \binom{n}{6} \left(\frac{1}{2}\right)^n\)

4. **Using AP Condition**:

• Plugging into the AP condition, we get:\(2\binom{n}{5} = \binom{n}{4} + \binom{n}{6}\)

5. **Binomial Coefficients**:

• Using the property of binomial coefficients:\(\binom{n}{k} = \binom{n}{n-k}\).

6. **Finding Suitable \( n \)**:

• From the equation \(2\binom{n}{5} = \binom{n}{4} + \binom{n}{6}\), we substitute \( n = 7 \):
• \(\binom{7}{5} = \binom{7}{2} = 21\)
• \(2 \times \binom{7}{5} = 2 \times 21 = 42\)
• \(\binom{7}{4} = \binom{7}{3} = 35\) and \(\binom{7}{6} = \binom{7}{1} = 7\)
• Thus, \(35 + 7 = 42\), so \( n = 7 \) satisfies the condition.

7. **Verification for \( n = 14 \)**:

• Similarly, substitute \( n = 14 \) in the equation to verify the arithmetic progression condition. The calculation confirms that \( n = 14 \) also satisfies the condition.

Therefore, the values of \( n \) that satisfy the given condition are 7 and 14.