The probabilities that Kofi, Kwasi and Ama will pass a certain examination are (frac{9}{10}, frac{4}{5}) and x respectively. If the probability that only one of them will pass the examination is (frac{9}{50}), find the :
(a) value of x ;
(b) probability that at least one of them will pass the examination.
Explanation
(a) p(Kofi passes) = (frac{9}{10}), p(Kofi fails) = (frac{1}{10})
p(Kwasi passes) = (frac{4}{5}), p(Kwasi fails) = (frac{1}{5})
p(Ama passes) = x, p(Ama fails) = 1 – x.
p(only one passes) = p(only Kofi passes) + p(only Kwasi passes) + p(only Ama passes)
= ((frac{9}{10} times frac{1}{5} times (1 – x)) + (frac{4}{5} times frac{1}{10} times (1 – x)) + (x times frac{1}{10} times frac{1}{5}))
= (frac{9(1 – x)}{50} + frac{4(1 – x)}{50} + frac{x}{50})
(implies frac{9}{50} = frac{9 – 9x + 4 – 4x + x}{50} = frac{13 – 12x}{50})
(therefore 13 – 12x = 9 implies 12x = 4)
(x = frac{4}{12} = frac{1}{3}).
(b) p(at least one passes) = 1 – p(none passes).
p(none passes) = (frac{1}{10} times frac{1}{5} times frac{2}{3})
= (frac{1}{75})
p(at least one passes) = (1 – frac{1}{75})
= (frac{74}{75})