Question

whenever horses a,b and c race together , their respective probabilities of winning are 0.3, 0.5 and 0.2. they race three times .

(a) Find the probability that the same horse wins all three races .

(b) Find the probability that a,b,,c each win one race.

Answer 1

Given $P(a)=0.3, P(b)=0.5, P(c)=0.2$

(a)

$$P(aaa)=(P(a))^3=(0.3)^3=0.027$$

$$P(bbb)=(P(b))^3=(0.5)^3=0.125$$

$$P(ccc)=(P(c))^3=(0.2)^3=0.008$$

$$P(one\ wins\ all)=0.027+0.125+0.008=0.16$$

(b)

First race $a$, second race $b$, third race $c$

$$P(abc)=0.3(0.5)(0.2)=0.03$$

First race $a$, second race $c$, third race $b$

$$P(abc)=0.3(0.2)(0.5)=0.03$$

First race $b$, second race $a$, third race $c$

$$P(bac)=0.5(0.3)(0.2)=0.03$$

First race $b$, second race $c$, third race $a$

$$P(bca)=0.5(0.2)(0.3)=0.03$$

First race $c$, second race $a$, third race $b$

$$P(cab)=0.2(0.3)(0.5)=0.03$$

First race $c$, second race $b$, third race $a$

$$P(cba)=0.2(0.5)(0.3)=0.03$$

$$P(each\ wins\ one\ race)=6(0.03)=0.18$$