Question

$X$ follows a binomial distribution with parameters $n=6$ and $p$. If $4P(X=4)=P(X=2)$, then $p$ is:

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{6}$
• D. $\frac{1}{3}$

Answer 1

The correct option is D $\frac{1}{3}$

For binomial distribution, $P(X=r)={}^n{C}_r{p}^r(1-p{)}^{n-r}$. Here, n =6.

According to question, $4P(X=4)=P(X=2)$

$\Rightarrow 4\times {}^6{C}_4{p}^4(1-p{)}^2={}^6{C}_2{p}^2(1-p{)}^4$

$\Rightarrow 4\times 15p^4(1-p{)}^2=15p^2(1-p{)}^4$

$\Rightarrow 4p^2=(1-p{)}^2\Rightarrow 3p^2+2p-1=0$

$\Rightarrow 3p^2+3p-p-1=0\Rightarrow 3p(p+1)-1(p+1)=0$

$\Rightarrow (p+1)(3p-1)=0$

$\therefore$ Either $p=-1$ or $p=\frac{1}{3}$

$\because 0\le p\le 1$

$\therefore p=\frac{1}{3}$