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, and the probability that a, b, c each wins one race are, respectively

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, and the probability that a, b, c each wins one race are, respectively

(A) 8/50, 9/50

(B) 16/100, 3/100

(C) 12/50, 15/50

(D) 10/50, 8/50

Answer : option (A)

Solution :

P(a) = 0.3, P(b) = 0.5 and P(c) = 0.2 , hence a, b and c are exhaustive

P(same horse will win all the three races) = P(aaa or bbb or ccc)

\begin{aligned}=\left( 0.3\right) ^{3}+\left( 0.5\right) ^{3}+\left( 0.2\right) ^{3}\ =\dfrac{27+125+8}{1000}\ =\dfrac{160}{1000}\ =\dfrac{4}{25}\end{aligned}

P(each horse will win exactly one race) = P(abc or acb or bca or bac or cab or cba)

$\Rightarrow 0.3\times 0.5\times 0.2\times 6=0.18=\dfrac{9}{50}$
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