If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X …

Question

If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then $${{P(X = 2)} \over {P(X = 3)}}$$ is equal to :

Accepted Answer

Given,

$n = 5$

and $P(X = 0) = P(X = 1)$

We know, $$p(x = r) = {}^n{C_r}\,.\,{p^r}\,.\,{q^{n - r}}$$

where $p + q = 1$

$$\therefore$$ $P(X = 0) = P(X = 1)$

$$ \Rightarrow {}^5{C_0}\,.\,{p^0}\,.\,{q^5} = {}^5{C_1}\,.\,{p^1}\,.\,{q^4}$$

$$ \Rightarrow 1\,.\,1\,.\,{(1 - p)^5} = 5\,.\,p\,.\,{(1 - p)^4}$$

$$ \Rightarrow 1 - p = 5p$$

$$ \Rightarrow 6p = 1$$

$$ \Rightarrow p = {1 \over 6}$$

$$\therefore$$ $$q = 1 - p = 1 - {1 \over 6} = {5 \over 6}$$

Now, $${{P(X = 2)} \over {P(X = 3)}} = {{{}^5{C_2}\,.\,{p^2}\,.\,{q^3}} \over {{}^5{C_3}\,.\,{p^3}\,.\,{q^2}}}$$

$$ = {{10\,.\,q} \over {10\,.\,p}}$$

$$ = {5 \over 6} \times {6 \over 1}$$

$= 5$

Solutions · 1