MCQ
If $X$ has a binomial distribution, $B( n, p)$ with parameters $n$ and $p$ such that $P(X\, = 2)\, = P (X\, = 3)$, then $E(X)$, the mean of variable $X$, is
- A$2 - p$
- ✓$3 - p$
- C$\frac{p}{2}$
- D$\frac{p}{3}$
$p)$
$\therefore \mathrm{P}(X=2)=^{n} \mathrm{C}_{2}(p)^{2}(1-p)^{n-2}$
and $P(X=3)=^{n} C_{3}(p)^{3}(1-p)^{n-3}$
Given $\mathrm{P}(X=2)=\mathrm{P}(X=3)$
$\Rightarrow^{n} \mathrm{C}_{2} p^{2}(1-p)^{n-2}=^{n} \mathrm{C}_{3}(p)^{3}(1-p)^{n-3}$
$ \Rightarrow \frac{{n!}}{{2!(n - 2)!}}.\frac{{{p^2}{{(1 - p)}^n}}}{{{{\left( {1 - p} \right)}^2}}}$
$ = \frac{{n!}}{{3!(n - 3)!}} \cdot \frac{{{p^3}{{(1 - p)}^n}}}{{{{\left( {1 - p} \right)}^3}}}$
$\Rightarrow \frac{1}{n-2}=\frac{1}{3} \cdot \frac{p}{1-p}$
$\Rightarrow 3(1-p)=p(n-2)$
$\Rightarrow 3-3 p=n p-2 p$
$\Rightarrow n p=3-p$
$\Rightarrow \mathrm{E}(X)=\mathrm{m} \operatorname{ean}=3-p$
$(\because \operatorname{mean} \text { of } \mathrm{B}(n, p)=n p)$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.