Question

A bag contains 6 R, 4 W and 8 B balls. If 3 balls are drawn simultaneously at random determine the probability of the event
(1) all 3 are red .
(2) all 3 are black
(3) at least 1 is red
(4) 1 of each colour are drawn
(5) the balls are drawn in the order of red, white, black.
(6) 2 are white and 1 is red.

Answer 1

i) P(all red) $=\frac{{}^6{C}_3}{{}^{18}{C}_3}=\frac{\frac{\mathrm{6.5.4}}{3!}}{\frac{\mathrm{18.17.16}}{3!}}=\frac{5}{204}$

ii) P(all black) $=\frac{{}^8{C}_3}{{}^{18}{C}_3}=\frac{\mathrm{8.7.6}}{\mathrm{18.17.16}}=\frac{7}{102}$

iii) P(at least 1 red) $=\frac{{}^6{C}_1.{}^{12}{C}_2}{{}^{18}{C}_3}+\frac{{}^6{C}_2{}^{12}{C}_1}{{}^{18}{C}_3}+\frac{{}^6{C}_3{}^{12}{C}_0}{{}^{18}{C}_3}$

$=\frac{6\times 66}{\frac{\mathrm{18.17.16}}{\mathrm{3.2.1}}}+\frac{15\times 12}{\frac{\mathrm{18.17.16}}{\mathrm{3.2.1}}}+\frac{5}{204}$

$=\frac{396}{816}+\frac{15}{68}+\frac{5}{204}$

$=\frac{396}{816}+\frac{15\times 12}{68\times 12}+\frac{5\times 4}{204\times 4}$

$=\frac{596}{816}=\frac{149}{204}$

iv) P(1 of each color) $=\frac{{}^4{C}_1{}^6{C}_1{}^8{C}_1}{{}^{18}{C}_3}=\frac{4\times 6\times 8}{\frac{\mathrm{18.17.16}}{\mathrm{3.2.1}}}$

$=\frac{4\times 6\times 8\times 3\times 2}{18\times 17\times 16}=\frac{4}{17}$

v) P(RWB) $=\frac{{}^4{C}_1{}^6{C}_1{}^8{C}_1}{{}^{18}{C}_1{}^{17}{C}_1{}^{16}{C}_1}=\frac{4\times 6\times 8}{18\times 17\times 16}=\frac{2}{51}$

vi) P(2W1R) $=\frac{{}^4{C}_2{}^6{C}_1}{{}^{18}{C}_3}=\frac{6\times 6}{\frac{18\times 17\times 16}{3.2}}=\frac{6\times 6\times 3\times 2}{18\times 17\times 16}$

$=\frac{3}{68}$