Question

The probability function of a binomial distribution is $P\left(x\right)={{}^{6}C}_x{p}^x{q}^{6-x},x=0,1,2,...,6$. If $2P\left(2\right)=3P\left(3\right)$, then $p=$____

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{5}$

Answer 1

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

$P\left(x\right)={{}^{6}C}_x{p}^x{q}^{6-x},x=0,1,2,...,6$

Given $2P\left(2\right)=3P\left(3\right)$
$\Rightarrow 2\times {{}^{6}C}_2{p}^2{q}^4=3\times {{}^{6}C}_3{p}^3{q}^3$
$\Rightarrow 2\left(\frac{6!}{4!2!}{p}^2{q}^4\right)=3\left(\frac{6!}{3!3!}{p}^3{q}^3\right)$
$\Rightarrow 30p^2{q}^4=60p^3{q}^3$
$\Rightarrow \frac{2p}{q}=1$
$\Rightarrow 1-p=2p$
$\Rightarrow p=\frac{1}{3}$