Question

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

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

Also, $2P\left(2\right)=3P\left(3\right)$

$2\times {}^6{C}_2{p}^2{q}^4=3\times {}^6{C}_3{p}^3{q}^3$

$\Rightarrow 2\times \frac{6!}{2!4!}{p}^2{q}^4=3\times \frac{6!}{3!3!}{p}^3{q}^3$

$\Rightarrow q=2p$
(as $p+q=1)$.

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