Simple probability problem

You're taking a final in which the professors review sheet says that a part of the final will be 2 comparative essays. You are given 8 possible writers that we talked about in a certain time frame, and you will have to compare 2 sets of them (4 writers total). There are 10 writers in said time frame, so he will be dropping 2 writers.

If you write 2 comparisons the day before the final, what is the probability that one or more of your chosen writers will be dropped from the list?

You're taking a final in which the professors review sheet says that a part of the final will be 2 comparative essays. You are given 8 possible writers that we talked about in a certain time frame, and you will have to compare 2 sets of them (4 writers total). There are 10 writers in said time frame, so he will be dropping 2 writers.

If you write 2 comparisons the day before the final, what is the probability that one or more of your chosen writers will be dropped from the list?

It looks like a straight-forward 50%, but I think I may be wrong.

Pick 2 from 10 = 45 different ways
Pick 2 from 6 = 15 different ways

So 15 of 45 possible sets of 8 authors contain all 4 you want

ie 2/3 chance one or more will be dropped.


Working from that original set of 45 you can also figure out which ones remove 2 authors you chose (2 of 4 = 6) of which 2/3 of combinations make both your essays useless.

So it's 41/45 that you'll get at least one useful essay, 1/3 that you'll get 2.

Pick 2 from 10 = 45 different ways
Pick 2 from 6 = 15 different ways

So 15 of 45 possible sets of 8 authors contain all 4 you want

ie 2/3 chance one or more will be dropped.


Working from that original set of 45 you can also figure out which ones remove 2 authors you chose (2 of 4 = 6) of which 2/3 of combinations make both your essays useless.

So it's 41/45 that you'll get at least one useful essay, 1/3 that you'll get 2.

Damn.

Thanks a lot.