Slot method : If a jury of 12 people is to be selected rando

[leungp2]

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?

24/91
45/91
2/3
67/91
84/91

Can you solve this problem using only Slot Method? Particularly, can you use one formula to get "the total number of juries that could be randomly selected from this jury pool"? I understand that you can add up each of the possible combinations using slot method (7M5W, 8M4W, etc) but I am looking for a faster way.

Thanks!

[RonPurewal]

I understand that you can add up each of the possible combinations using slot method (7M5W, 8M4W, etc) but I am looking for a faster way.

Thanks!

the second method listed below may qualify as a "faster method" -- but the much more important point is that you don't really need a faster method. the method you've mentioned here is quite efficient already.

what you should NEVER do, during the actual exam, is sit on a perfectly good solution -- something that you know will solve the problem -- and not use it, because you're thinking about whether there's a "faster" solution. if you can think of something that will work -- or even something that might, maybe, work -- then do it.

--

if you want to calculate the probability directly, there are only three possible cases:
8m 4w (2/3 of 12 = 8 is the minimum number of men)
9m 3w
10m 2w
that's all, since there are only 10 men in the pool.
just calculate these three, and add them up.

you can also calculate the probability indirectly, by finding the cases you DON'T want...
7m 5w
... that's all the cases you don't want (since it's not possible to select more than 5 women).
so just find the probability of getting 7 men and 5 women, and subtract it from 1.

--

there's only one thing i don't understand:
in your solution, you have "7m 5w" and "8m 4w" and then "etc." -- implying that the "7m 5w" and the "8m 4w" are part of the same solution process.
that doesn't make sense; those two should be mutually exclusive. i.e., if you're calculating the probability directly, then you should ignore 7m/5w and use 8m/4w (as well as the cases with more men); if you're calculating it indirectly, you should use 7m/5w and ignore 8m/4w.
how would you use both of those?