(a) The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins : (i) all races ; (ii) no race ; (iii) at most 3 races ? (b) A class consists of 5 girls and 10 boys. If a committee of 5 is chosen at random from the class, find the probability that : (i) 3 boys are selected ; (ii) at least one girl is selected.
Explanation
(a) p(man wins) = p = 0.8 ; p(man loses) = q = 0.2. 4 different races : \((p + q)^{4} = p^{4} + 4p^{3}q + 6p^{2} q^{2} + 4pq^{3} + q^{4}\) (i) p(man wins all) = \(p^{4} = (0.8)^{4} = 0.4096\) (ii) p(man wins no race) = \(q^{4} = (0.2)^{4} = 0.0016\) (iii) p(man wins at most 3) = 1 - p(man wins all) = 1 - 0.4096 = 0.5904. (b)(i) 5 girls, 10 boys ; Total = 15. Ways of selecting 3 boys = \(^{10}C_{3} \times ^{5}C_{2}\) = \(120 times 10\) = 1200 Without restriction, selection = \(^{15}C_{5}\) = \(\frac{15!}{10! 5!}\) = 3003 p(selecting 3 boys) = \(\frac{1200}{3003} = 0.3996\) (ii) Ways of selecting no girls = \(^{10}C_{5} \times ^{5}C_{0}\) = 252 p(at least one girl) = 1 - p(no girl) = \(1 - \frac{252}{3003}\) = \(2751}{3003} = 0.9161\)
