So, the question is:
There are 7 students in a class: 2 boys and 5 girls.
If the teacher picks a group of 3 at random, what is the probability that everyone in the group is a girl?
Actually, I know how to solve this problem using combination: $\frac{\text{# of ways to choose 3 girls out of 5}}{\text{total number of ways to choose 3 students out of 7}}$ which yields $\frac{2}{7}$. I wanted to solve this problem in a different way but failed and want to know why? So, what I tried to do was to find probability of getting one group of 3 girls, GGG, which is $(\frac{5}{7})^3$. Then multiplied it with the # of ways to choose 3 girls out of 5, which is 10, and got totally different result, 3.6 .
So what is wrong with the second approach?
P.S. It is the first time I am asking a question here and therefore i am sorry if there is any problem with the post overall.
2026-03-29 02:10:50.1774750250
Finding probability with combination
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The answer is: 5/7 * 4/6 * 3/5 which makes 2/7. (5/7)^3 is wrong because, as lulu pointed out, choosing a girl first lowers the probability that the second is also a girl. Multiplying by 10 is also wrong because there is only one unique collection of a group of 3 girls, GGG.