Five students are to be selected from a large population. If 60% of them are boys and the rest are girls, find the probability that :
(a) exactly 3 of them are boys;
(b) at least 3 of them are girls.
Explanation
60% boys, 40% girls.
p(boy) = p = 60% = 0.6
p(girl) = q = 40% = 0.4
The binomial distribution function is
((p + q)^{5} = p^{5} + 5p^{4} q + 10p^{3} q^{2} + 10p^{2} q^{3} + 5pq^{4} + q^{5})
(a) p(exactly 3 boys) = (10p^{3} q^{2})
(10(frac{3}{5})^{3} (frac{2}{5})^{2})
= (10(frac{27}{125})(frac{4}{25}))
= (frac{1080}{3125})
= (frac{216}{625})
(b) p(at least 3 are girls) = 1 – [p(0 girls) + p(1 girl) + p(2 girls)].
= (1 – [p^{5} + 5p^{4}q + 10p^{3} q^{2}])
= (1 – [(frac{3}{5})^{5} + 5(frac{3}{5})^{4} (frac{2}{5}) + 10(frac{3}{5})^{3} (frac{2}{5})^{2}])
= (1 – [frac{243}{3125} + frac{810}{3125} + frac{1080}{3125}])
= (1 – [frac{2133}{3125}])
= (frac{992}{3125}).