A survey conducted revealed that four out of every twenty taxi drivers do not have...

Explanation

p(have valid license) = p = 0.8

p(no valid license) = q = 0.2.

The binomial probability distribution is

\((p + q)^{6} = p^{6} + 6p^{5} q + 15p^{4} q^{2} + 20p^{3} q^{3} + 15p^{2} q^{4} + 6p q^{5} + q^{6}\)

(a) p(2 have valid licenses) = \(15p^{2} q^{4}\)

= \(15 (0.8)^{2} (0.2)^{4} = 15(0.64)(0.0016)\)

= \(0.01536 \approxeq 0.015\) (3 d.p)

(b) p(more than 3 have valid licenses) = \(p^{6} + 6p^{5} q + 15p^{4} q^{2}\)

= \((0.8)^{6} + 6(0.8)^{5} (0.2) + 15(0.8)^{4} (0.2)^{2}\)

= \(0.262144 + 0.393216 + 0.24578\)

= \(0.90112 \approxeq 0.901\) (3 d.p)

(c) p(at least 5 have valid licenses) = \(p^{6} + 6p^{5} q\)

= \((0.8)^{6} + 6(0.8)^{5} (0.2)\)

= \(0.262144 + 0.393216\)

= \(0.65536 \approxeq 0.655\).