Question

The random variable \(X \sim \text{Bin}(11,p)\). If \(P(X=8)=P(X=7)\), then \(p=\)

• A. \(\frac{2}{3}\)
• B. \(\frac{1}{3}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{1}{4}\)

Hint:

For equality of consecutive binomial terms: \[ \boxed{ \frac{P(r+1)}{P(r)}=\frac{n-r}{r+1}\cdot\frac{p}{1-p} } \] Use ratio method to solve faster in exams.

Answer 1

The Correct Option is C

Concept: For a binomial distribution, \[ P(X=r)=\binom{n}{r}p^r(1-p)^{n-r} \] Given: \[ X \sim \text{Bin}(11,p), \quad P(X=8)=P(X=7) \]

Step 1: Write both probabilities.
\[ P(X=8)=\binom{11}{8}p^8(1-p)^3 \] \[ P(X=7)=\binom{11}{7}p^7(1-p)^4 \]

Step 2: Equate and simplify.
\[ \binom{11}{8}p^8(1-p)^3=\binom{11}{7}p^7(1-p)^4 \] Cancel common terms: \[ \binom{11}{8}p = \binom{11}{7}(1-p) \] We know: \[ \binom{11}{8}=\binom{11}{3}=165,\quad \binom{11}{7}=\binom{11}{4}=330 \] So: \[ 165p = 330(1-p) \]

Step 3: Solve for \(p\).
\[ 165p = 330 - 330p \] \[ 495p = 330 \] \[ p = \frac{330}{495} = \frac{2}{3} \]

Step 4: Choose correct option.
\[ \boxed{p=\frac{2}{3}} \Rightarrow Option (A) \]