Probability Problem #1

A jury pool contains 10 men and 5 women.  If the 12 jurors are randomly selected from this pool, what is the probability that the jury is comprised of ²⁄₃ men?

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When you are doing a probability problem and the math looks a little too complicated to finish in a reasonable amount of time, check to see if the opposite of what you're trying to figure out looks easier.  In this case, figuring out the chances that 8 of the 12 jurors selected are men might take a while, but notice that the only way for the fraction of the jury being male to be less than ²⁄₃ is if all three of the non-selected people are male (so you end up with seven male jurors and all five of the female jurors).  That's a much easier problem:

For the first of the three selections for people not on the jury to be a man, the probability is 10 out of 15. For the second, there are 14 people to choose from and 9 men, so 9 out of 14. Similarly, the last is 8 out of 13.

¹⁰⁄₁₅ * ⁹⁄₁₄ * ⁸⁄₁₃ =

^{10 2}⁄_{15 3} * ⁹⁄₁₄ * ⁸⁄₁₃ =

^{2 1}⁄₃ * ⁹⁄_{14 7} * ⁸⁄₁₃ =

¹⁄_{3 1} * ^{9 3}⁄₇ * ⁸⁄₁₃ =

¹⁄₁ * ³⁄₇ * ⁸⁄₁₃ = ²⁴⁄₉₁

And since this is the possibility that we'll have all men in the group not selected, which is the opposite of what we want to know about, the actual answer is 1 - ²⁴⁄₉₁ or ⁶⁷⁄₉₁.