Question

In a hurdle race, a player has to cross $5$ hurdles. The probability that he will clear each hurdle is $\frac{2}{3}$. The probabilty that he will knock down fewer than $2$ hurdles is

$\left(a\right)\frac{32}{243}\left(b\right)\frac{80}{243}$
$\left(c\right)\frac{112}{243}\left(d\right)\frac{122}{243}$

Answer 1

Probability of clearing each hurdle, $q=\frac{2}{3}$
Probability that he will knock down the hurdle, $p=1-\frac{2}{3}=\frac{1}{3}$

By using Bernoulli trial, the probabilty that he will knock down fewer than $2$ hurdles is,
$P(X=0)+P(X=1)={}^n{C}_0{p}^0{q}^{n-0}+{}^n{C}_1{p}^1{q}^{n-1}$
$={}^5{C}_0{\left(\frac{2}{3}\right)}^5+{}^5{C}_1\left(\frac{1}{3}\right){\left(\frac{2}{3}\right)}^4$
$=\frac{32}{243}+5\times \frac{16}{243}$
$=\frac{112}{243}$

Answer: $\left(c\right)$