Question

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trails, there will be at least 3 successes.

Answer 1

If $p$ is the probability of a success and $q$, that of a failure,then $p+q=1$ and $p=3q$

$\Rightarrow 3q+q=1\Rightarrow q=\frac{1}{4},p=1-\frac{1}{4}=\frac{3}{4}$

$\text{Here number of trails}=5$

$\text{By binomial distribution, we have}$
$P(x=r)={}^n{C}_r.p^r.q^{n-r}$
$\text{Now,}$
$\text{P(getting at least 3 success)=P(X=3)+P(X=4)+P(X=5)}$
$={}^5{C}_3.p^3.q^2+{}^5{C}_4.p^4.q^1+{}^5{C}_5.p^5.q^0$
$={}^5{C}_3.(\frac{3}{4}{)}^3.(\frac{1}{4}{)}^2+{}^5{C}_4.(\frac{3}{4}{)}^4.(\frac{1}{4}{)}^1+{}^5{C}_5.(\frac{3}{4}{)}^5.(\frac{1}{4}{)}^0$
$=(\frac{3}{4}{)}^3\left[\begin{array}{c}{}^5{C}_3.(\frac{1}{4}{)}^2+{}^5{C}_4.(\frac{3}{4}{)}^1.(\frac{1}{4}{)}^1+{}^5{C}_5.(\frac{3}{4}{)}^2\end{array}\right]$
$=\frac{27}{64}\left[\begin{array}{c}\frac{10}{16}+\frac{15}{16}+\frac{9}{16}\end{array}\right]$
$=\frac{27}{64}\times \frac{34}{16}=\frac{459}{512}$