Question

If a random variable X follows the Binomial distribution \(B(10, \quad p)\) such that \(5 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)\), then the value of \(\frac{\mathrm{P}(\mathrm{X}=5)}{\mathrm{P}(\mathrm{X}=6)}\) is equal to

• A. \(\frac{6}{5}\)
• B. \(\frac{2}{5}\)
• C. \(\frac{12}{5}\)
• D. \(\frac{1}{5}\)

Hint:

Use \(5P(X=0)=P(X=1)\) to find p, then form the ratio of the two terms.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For \(B(n,p)\), \(P(X=r)={}^nC_r\,p^rq^{n-r}\) with \(q=1-p\).

Step 2: Find p
\(5q^{10}=10pq^9\Rightarrow q=2p\). Since \(p+q=1\), \(p=\dfrac{1}{3}\) and \(q=\dfrac{2}{3}\).

Step 3: Form the Ratio
\(\dfrac{P(5)}{P(6)}=\dfrac{{}^{10}C_5\,p^5q^5}{{}^{10}C_6\,p^6q^4}=\dfrac{252}{210}\cdot\dfrac{q}{p}\)

Step 4: Calculation
\(=\dfrac{6}{5}\times2=\dfrac{12}{5}\)

Final Answer:
The value of the ratio is 12/5. \[ \boxed{\text{(C) }\dfrac{12}{5}} \]