question_answer

In a hurdles race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is \[\frac{5}{6}.\] What is the probability that he will knock down fewer than 2 hurdles?

Answer:

Given, total number of hurdles = 10 Probability of clearing each hurdle \[=\frac{5}{6}\] \[\therefore \]Probability of knocking a hurdle \[=1-\frac{5}{6}=\frac{1}{6}\] \[\therefore \]Probability that he will knock down fewer than 2 hurdles = P (knocking 0 hurdle) + P (knocking 1 hurdle) \[={}^{10}{{C}_{0}}{{\left( \frac{1}{6} \right)}^{0}}{{\left( \frac{5}{6} \right)}^{10}}+{}^{10}{{C}_{1}}{{\left( \frac{1}{6} \right)}^{1}}{{\left( \frac{5}{6} \right)}^{9}}\] \[=1\times 1\times {{\left( \frac{5}{6} \right)}^{10}}+10\times \frac{{{5}^{9}}}{{{6}^{10}}}\] \[={{\left( \frac{5}{6} \right)}^{10}}+2\left( \frac{{{5}^{10}}}{{{6}^{10}}} \right)=3{{\left( \frac{5}{6} \right)}^{10}}=\frac{{{5}^{10}}}{2\times {{6}^{9}}}\]