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})