Question

Two horses are considered for a race. The probability of selection of the first horse is 1/4 and that of the second is 1/3. 

What is the probability that:

(a) both of them will be selected.

(b) only one of them will be selected.

(c) none of them will be selected.

Answer 1

Let the probability of selecting first horse = P(A) = 1/4

and the probability of selecting second horse = P(B) = 1/3

then, P(\(\bar{A}\)) = 1 − P(A) = 1 - \(\frac{1}{4}=\frac{3}{4}\)

and P(\(\bar{B}\)) = 1 - P(B) = 1 - \(\frac{1}{3}=\frac{2}{3}\)

(a) P(Both of them selected) 

= P(A). P(B)

\(=\frac{1}{4}\times\frac{1}{3}=\frac{1}{12}\)

(b) P(Only one selected)

\(=P(A)\,P(\bar{B})+P(\bar{A})\,P(B)\)

\(=\frac{1}{4}\times\frac{2}{3}+\frac{1}{3}\times\frac{3}{4}=\frac{5}{12}\)

(c) P(None of them selected)

\(=P(\bar{A})\,P(\bar{B})\)

\(=\frac{3}{4}\times\frac{2}{3}\)

\(=\frac{1}{2}\)