A bag contains 4 red, 6 blue and 8 green identical marbles. (a) If three
A bag contains 4 red, 6 blue and 8 green identical marbles.
(a) If three marbles are drawn at random, without replacement, calculate the probability that :
(i) all will be green ; (ii) all will have the same colour.
(b) If each marble is replaced before another is drawn, calculate the probability that all will have the same colour.
Explanation
4 red, 6 blue, 8 green = 18 marbles.
p(red) = \(\frac{4}{18} = \frac{2}{9}\); p(not red) = \(\frac{7}{9}\)
p(blue) = \(\frac{6}{18} = \frac{1}{3}\) ; p(not blue) = \(\frac{2}{3}\)
p(green) = \(\frac{8}{18} = \frac{4}{9}\) ; p(not green) = \(\frac{5}{9}\)
(a) p( 3 green) = \(\frac{8}{18} \times \frac{7}{17} \times \frac{6}{16} = \frac{7}{102}\)
(b) p(all same colour) = p(all red) or p(all blue) or p(all green)
= \(\frac{4}{18} \times \frac{4}{18} \times \frac{4}{18} + \frac{6}{18} \times \frac{6}{18} \times \frac{6}{18} + \frac{8}{18} \times \frac{8}{18} \times \frac{8}{18}\)
= \(\frac{792}{5832}\)
= \(\frac{11}{81}\)