Question

In a binomial distribution \[ P(X=2)\div P(X=23)=\left(\frac{2}{3}\right)^{21}, \] then the mean of the binomial distribution is

• A. \(10\)
• B. \(15\)
• C. \(20\)
• D. \(25\) \bigskip

Hint:

For binomial distributions, whenever probabilities of two terms are compared, first use \[ P(X=r)=\binom{n}{r}p^rq^{\,n-r}. \] Look for complementary indices since \[ \binom{n}{r}=\binom{n}{n-r}, \] which often helps determine \(n\) immediately.

Answer 1

The Correct Option is B

Concept: For a binomial distribution, ratio of probabilities simplifies to \(\frac{\binom{n}{2}}{\binom{n}{23}}(q/p)^{21}\). Use symmetry \(\binom{n}{r}=\binom{n}{n-r}\) to find \(n\), then solve for \(p\) and compute mean \(np\).

Step 1: Set up the ratio. \(\frac{\binom{n}{2}p^2q^{n-2}}{\binom{n}{23}p^{23}q^{n-23}} = \frac{\binom{n}{2}}{\binom{n}{23}}\left(\frac{q}{p}\right)^{21} = \left(\frac23\right)^{21}\).

Step 2: For coefficients to cancel, \(\binom{n}{2}=\binom{n}{23} \Rightarrow 23 = n-2 \Rightarrow n=25\).

Step 3: Then \(\frac{q}{p} = \frac23\). With \(p+q=1\), \(p + \frac23p = 1 \Rightarrow \frac53p=1 \Rightarrow p=\frac35\).

Step 4: Mean = \(np = 25 \times \frac35 = 15\).

Step 5: Write the final answer. \(\boxed{15}\)