I'm sure that there is a faster way to solve the problem but one solution is through combinations.
the pool consists of 15 people, 10 men (2/3) and 5 women (1/3).
we have to calculate the probability that the jury will comprise at least 2/3 of men, in order words at least 8 men. This includes a jury with 8 men, 9 men and 10 men.
first ways to select 12 people among 15 = C(15,12) = 15x14x13/3x2 = 5x7x13.
ways to select 8 men among 10 and 4 women among 5 = C(10,8) x (5,4) = (10 x 9)/2 x 5 = 9 x 5 x 5
ways to select 9 men among 10 and 3 women among 5 = C(10,9) x C(5,3) = 10 x 10
ways to select 10 men among 10 and 2 women among 5 = C(10,10) x C(5,2) = 1 x 10
probability = desire possibilities / total possibilities =
((9 x 5 x 5) + (10 x 10) + (10))/5x7x13 = ( 9x5 + 2x10 +5)/7x13 = 67/91
