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.

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 $a$

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

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