(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)