Question

In a race, the odds in favour of horses A, B, C, D are 1 : 3, 1 : 4, 1 : 5 and

1 : 6 respectively. Find probability that one of them wins the race.

Answer 1

We have,

$P\left(A\right):P\left(\overline{A}\right)=1:3$

$\Rightarrow P\left(A\right)=\frac{1}{4}$

$P\left(B\right):P\left(\overline{B}\right)=1:4$

$\Rightarrow P\left(B\right)=\frac{1}{5}$

$P\left(C\right):P\left(\overline{CB}\right)=1:5$

$\Rightarrow P\left(C\right)=\frac{1}{6}$

$P\left(D\right):P\left(\overline{D}\right)=1:6$

$\Rightarrow P\left(D\right)=\frac{1}{7}$

$\therefore$ Probability that at least one of them wins is given by

$P(A\cup B\cup C\cup D)$

$=P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right)$

=$\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}=\frac{319}{420}$