If a jury of 12 people is to be selected randomly from a pool of 15 potential jurors, and the jury pool consists of 2/3 men and 1/3 women, what is the probability that the jury will comprise at least 2/3 men?

A) 24/91
B) 45/91
C) 2/3
D) 67/91
E) 84/91

Maybe my logic is off, or maybe I shouldn't have taken a practice test on a holiday, but I don't understand the solution, especially the part that says, "The easiest way to calculate this probability is to use the “1-x shortcut.” The only way the jury will have fewer than 8 men is if a jury of 7 men and 5 women (the maximum number of women available) is selected. There cannot be fewer than 7 men on the jury, since the jury must have 12 members and only 5 women are available to serve on the jury."

Since ther are 10 men and of the 12 jurors to be selected 8 wil have to be men ((2/3)*(12)=12), I did 10!/8!*(10-8)! and got (10*9)/2! =45. Then for the women, I did 5!/4!(5-4)! and got 5. Multiplied the two together to get 225.

225/445 [ I got 455 from 15!/12!(15-12!)] and chose 45/91.

Obviously, this was a trap and I fell for it. I still don't know what I did wrong. I kind of understand the only way to get less than 8 is to get 7 men... but not really.

Can you explain this to me? Pretend I am a 1st grader and start from the beginning please! Thank you!

looks like you're making a mistaken assumption here:


the problem asks for the probability that AT LEAST 8 jurors will be male. you have only calculated the probability that EXACTLY 8 jurors will be male.
in particular, if you're going to use this sort of direct approach, you have to consider the possibilities in which MORE than 8 jurors will be male: 9 male/3 female, or 10 male/2 female. (there's no need to consider 11+ male, because there are only 10 males in the pool.)




when you use the (1 - x) deal, you need to find the probability of the opposite event. since the event under consideration is "at least 8 males", the opposite event is "fewer than 8 males".
now technically, this would encompass all kinds of possibilities: 7m/5f, 6m/6f, 5m/7f, ..., all the way down to 0m/12f. but the only one of these that's possible in the first place is 7m/5f, because there are only 5 females in the pool in the first place. therefore, you don't consider the possibilities with 6+ females, because they're impossible.



that'd be one smart first-grader.
mm-hmm