Question

In a hurdle race , a runner has probability $p$ of jumping over a speciafic hurdle. given that in $5$ trials, the runner succeeded $3$ times , the conditional probability that the runner had succeeded in the first trial, is :

• A. $3/5$
• B. $2/5$
• C. $1/5$
• D. none of these

Answer 1

Answer: (A)

$3/5$

Let X denote the event that the runner succeed exactly $3$ times out of five trials and Y denotes the event the runner succeeds on the first trial then,

$P\left(\frac{Y}{X}\right)=\frac{P(Y\bigcap X)}{P\left(X\right)}$

$P(Y\bigcap X)$denotes the probability of succeeding in the first trial and exactly once in two other trials.

$P(Y\bigcap X)=p\left[{}^4{c}_2{p}^2\right(1-p{)}^2]=6p^3(1-p{)}^2$

and $P\left(X\right){=}^5{c}_3{p}^3(1-p{)}^2=10p^3(1-p{)}^2$

$P\left(\frac{B}{A}\right)=\frac{6p^3(1-p{)}^2}{10p^3(1-p{)}^2}$

$P\left(\frac{B}{A}\right)=\frac{3}{5}$

The conditional probability that the runner had succeeded in the first is $\frac{3}{5}$.