Question

An unbiased coin is tossed \(n\) times. If the probability of getting 5 heads is equal to the probability of getting 6 heads, then the probability of getting 3 heads is:

• A. \(^{11}C_5 \left(\frac{1}{2}\right)^5\)
• B. \(^{11}C_6 \left(\frac{1}{2}\right)^6\)
• C. \(^{11}C_3 \left(\frac{1}{2}\right)^{11}\)
• D. \( \frac{11}{1024} \)

Hint:

Equal binomial probabilities occur when \(r\) and \(n-r\) are equal → use symmetry.

Answer 1

The Correct Option is C

Concept: Binomial probability: \[ P(r) = {^nC_r}\left(\frac{1}{2}\right)^n \]

Step 1: Given \[ P(5) = P(6) \Rightarrow {^nC_5} = {^nC_6} \]

Step 2: Property \[ {^nC_r} = {^nC_{n-r}} \] \[ \Rightarrow 5 + 6 = n \Rightarrow n = 11 \]

Step 3: Required probability \[ P(3) = {^{11}C_3}\left(\frac{1}{2}\right)^{11} \] \[ \therefore \text{Answer = } {^{11}C_3}\left(\frac{1}{2}\right)^{11} \]