Explanation:
remember, a probability is always the ratio of
desired cases / totally possible cases
it starts with all the combinations of 4 out of 12 (= 8 + 4) choices
C(12, 4) = 12! / (4! × (12-4)!) = 12! / (4! × 8!) =
= 12×11×10×9 / 24 = 11×5×9 = 495
these are our totally possible outcomes.
the desired outcomes with only 4 men in the selected group out of 12 people is the same amount as the possible selections of 4 men out of the available 8 men :
C(8, 4) = 8! / (4! × (8-4)!) = 8! / (4! × 4!) =
= 8×7×6×5 / 24 = 2×7×5 = 70
any other combination of C(12, 4) must contain at least 1 woman.
so the probability of getting 4 men out of the random pull is
70/495 = 0.1414141414...
we could get that also by saying the probability to get a man on the first pull is (12 people in total, 8 "desired" period with the right gender)
8/12 = 2/3 = 0.666666666...
now, we have 7 men out of 11 total people for the second pull. the probability here is
7/11 = 0.636363636...
then we have 6 men out of 10 total people for the third pull. the probability here is
6/10 = 3/5 = 0.6
and lastly, we have 5 men out of total 9 people for the fourth pull. the probability is
5/9 = 0.555555555...
the probability to get 4 men out of the random pulling is the combination of these 4 individual probabilities :
2/3 × 7/11 × 3/5 × 5/9 = 210 / 1485 = 70/495 =
= 0.1414141414...
it is confirmed.