Question

It is given that the discrete random variable \[ X\sim B(n,p) \] and \[ P(X=2)=P(X=3). \] Then mean of \(X\) is

• A. \(2-p\)
• B. \(3-p\)
• C. \(p-2\)
• D. \(p-3\)

Hint:

For binomial distribution, \[ P(X=r)={}^{n}C_r p^r(1-p)^{n-r} \] and mean is always \[ np. \] Equating two probabilities often helps in directly obtaining \(np\).

Answer 1

The Correct Option is B

Step 1: Write $P(X=2)=P(X=3)$ using the binomial formula.
$P(X=r) = ^nC_r p^r(1-p)^{n-r}$. So: \[ ^nC_2 p^2(1-p)^{n-2} = ^nC_3 p^3(1-p)^{n-3} \]

Step 2: Simplify by dividing both sides by $p^2(1-p)^{n-3}$.
\[ ^nC_2(1-p) = ^nC_3 \cdot p \]

Step 3: Substitute the combination values.
$^nC_2 = \frac{n(n-1)}{2}$, $^nC_3 = \frac{n(n-1)(n-2)}{6}$. So: \[ \frac{n(n-1)}{2}(1-p) = \frac{n(n-1)(n-2)}{6}\cdot p \] Cancel $n(n-1)$ (assuming $n \geq 3$): $\frac{1-p}{2} = \frac{(n-2)p}{6} \Rightarrow 3(1-p)=(n-2)p$.

Step 4: Solve for $np$.
$3-3p = np-2p \Rightarrow 3-p = np$. So $np = 3-p$.

Step 5: Identify the mean.
The mean of $X \sim B(n,p)$ is $np$. We found $np = 3-p$.

Step 6: State the answer.
\[ \boxed{3-p} \]