Question

Whenever horses $a,b,c$ race together, their respective probabilities of winning the race are $0.3,0.5$ and $0.2$ respectively. If they race three times the probability that "the same horse wins all the three races" is $p$ and the probability that $a,b,c$ each wins one race is $q$, then the value of $\frac{p}{q}$ is (Assume no dead heat)

• A. $\frac{9}{8}$
• B. $\frac{8}{9}$
• C. $\frac{3}{2}$
• D. $\frac{2}{3}$

Answer 1

The correct option is B $\frac{8}{9}$

Given:
$P\left(a\right)=0.3$
$P\left(b\right)=0.5$
$P\left(c\right)=0.2$
Probability that "the same horse wins all the three races"
$=(0.3{)}^3+(0.5{)}^3+(0.2{)}^3=\frac{27}{1000}+\frac{125}{1000}+\frac{8}{1000}=\frac{160}{1000}=\frac{8}{50}=p$
Now, $a,b,c$ each wins one race
$\therefore$ Total ways of happening $=3!=6$ ways
$\therefore \text{required probability}=6\times \left(0.3\right)\left(0.5\right)\left(0.2\right)$
$=6\times \frac{30}{1000}=\frac{9}{50}=q$
$\therefore \frac{p}{q}=\frac{8/50}{9/50}=\frac{8}{9}$