I am having a really hard time understanding how my teacher came up with the answer. I have an online class for statistics and well it's not as easy as I thought it would have been. Well my teacher is not very good at explaining things. She gave us the answer but I am not sure how she came up with the answer. I am better at step by step solutions. So here's the question...

There are 5 chemistry professors and 6 physics instructors at a college. If a committee of four instructors is chosen, find the probability that at least one of them is a physics instructor?

My teacher showed it like this

5/11 * 4/10 * 3/9 * 2/8 = 1/66

Than 1-1/66 = 65/66

why do the fractions reduce by 1 on the top and bottom? Why do we multiply them too?

There is another question that I attempted but got wrong on my homework if it is easier to explain this one.

A carpool contains 3 kindergartners and 5 first graders. If two children are ill find the probability that at least one of them is a kindergartner?

The reason we subtract from 1 is because that is the easiest way to tackle a 'at least one'

problem. At least one means we have the bare minimum... at least. Well, we find the

probability of none and subtract from 1. That is the opposite of none.

The professor found the prob. Of no physics and subtracted from 1.

If there are no physics, then they must all be chemistry.

The first chemist is 5/11 and so forth.

After the first is chosen there are 4 left out of the 11 remaining, so 4/10.

They reduce by 1 because we are choosing 1 each time we select and not replacing them.

We could also do it this way:



We are choosing 4 of the 5 chemists in the numerator and choosing 4 overall out of the

11 in the denominator. Then, because there were no physicists chosen, we subtract from

1 to get 'at least one' physicist.

See a little better now?

The kindergarten problem is worked the same way.

I do understand the Physics and science professor problem but the carpool question keeps messing me up now. The answer my teacher gave me was 9/14. This is what I got...

5/8 X 4/7 X 3/6 X 2/5 = 120/1680 which reduces to 1/14

which makes my answer 13/14

If I came to the answer the teacher got I should have came to 600/1680=5/14

Than the answer would have been 9/14

I don't know if I am using the right numbers to answer it.

As before, we want the prob. That 'at least one' kindergartener is sick. Therefore, we find the prob. Of no kindergarteners and subtract from 1.

If there are no kinder sick, then we are choosing two of the first graders.

Choose first 'first grader': (5/8)

Choose second 'first grader': (4/7)

(5/8)(4/7)=5/14

Now, subtract from 1:

1-5/14=9/14

If you are having trouble understanding the 'at least' idea, think of 'at least one' as having the minimum amount or more.

What's the least you can have? One. At least one or more. That means you have something. What is the opposite of something? Nothing.

That is why we find the prob. Of nothing and subtract from one. That is all that's left.

Suppose it said 'at least 2'. Depending on how many there are, it may be easier to find the prob. Of 0 or 1 and subtract from 1.