Question

Let $X$ follows binomial distribution with parameters $n,p$. If $P(X=3)=P(X=5)$ and $p>\frac{1}{2}$, then

• A. $5\le n\le 7$
• B. $n\ge 15$
• C. $n\ge 8$
• D. $3\le n\le 7$

Answer 1

The correct option is A

$5\le n\le 7$

$P(X=3)=P(X=5)\Rightarrow {\left(\frac{q}{p}\right)}^2=\frac{{}^n{C}_5}{{}^n{C}_3}$

$\because \frac{q}{p}<1\Rightarrow \frac{{}^n{C}_5}{{}^n{C}_3}<1\Rightarrow (n-3)(n-4)<20$
$\Rightarrow n<8$ and $n\ge 5$
$\because n\in N$
$\therefore n\in [5,7]$