A class of 400 students having 220 women and 180 men. The class is divided into study groups of 10 students each. What is the probability that a certain group will have exactly 4 men? I have used the following two approaches:
Combination question: Choose 4 men of 180 and choose 6 women of 220 divided by the total ways of choosing 10 people from 400. Answer = 180C4 * 220C6 / 400C10 = 0.24102549843
Binomial distribution: Total trials (n) = 10, number of success (x) = 4 and probability of success (p) = 180/400 = 0.45. Note: Here success = picking a man. So, P(4) = 10C4 * 0.45^4 * (1-0.45)^(10-4) = 0.23836664662
Why are these two outputs not the same? If one of my approach is incorrect, then which approach have I made mistake in? Also, how to rectify that mistake if any?
Answer given by @lulu in the comments.
The combination answer is correct. The binomial distribution assumes that all the trials are independent of each other but that is not the case in this question. Once a man is picked in the first trial, the probability of picking the second man becomes 179/399.
So the probability of picking the first man = 180/400. Probability of pickign the second man = 179/399. This implies that the next trial is dependent on the past trials.
Thanks to @N. F. Taussig for pointing out the hypergeometric distribution to be used in such a case and not the Binomial distribution.