Question

If random variable ${\displaystyle x\sim b(n=5,P=\frac{1}{3})}$, then P(2 < X < 4) is equal to ______.

Options

• ${\displaystyle \frac{80}{243}}$

• ${\displaystyle \frac{40}{243}}$

• ${\displaystyle \frac{40}{343}}$

• ${\displaystyle \frac{80}{343}}$

Answer 1

If random variable ${\displaystyle x\sim b(n=5,P=\frac{1}{3})}$, then P(2 < X < 4) is equal to ${\displaystyle \underline{\frac{40}{243}}}$.

Explanation:

Since, X ∼ B = ${\displaystyle (n=5,p=\frac{1}{3})}$

∵ p + q = 1

${\displaystyle \Rightarrow \frac{1}{3}+q}$ = 1

${\displaystyle \Rightarrow }$ q = ${\displaystyle \frac{2}{3}}$

∴ p(2 < X < 4) = p(X = 3)

= ${\displaystyle {}^5{C}_3{\left(\frac{1}{3}\right)}^3{\left(\frac{2}{3}\right)}^2}$

= ${\displaystyle 10\times \frac{4}{3^5}}$

= ${\displaystyle \frac{40}{243}}$