Balls in hat probability question

[tegan]

I've made it all the way to my senior year in college without having to struggle through math, so hopefully someone can help me with this basic probability problem!

A box contains one yellow, two red, and three green balls. Two balls are randomly chosen without replacement. Define the following events:
A: {One of the balls is yellow}
B: {At least one ball is red}
C: {Both balls are green}
D: {Both balls are of the same color}

Find the following conditional probabilities:

P(A|B) =
P(D|B^compliment) =
P(D|C^compliment) =

I am using a program called WebWork to do these problems, so I know right away if an answer I enter is correct, and so far everything I have tried has been incorrect.
What I have been working on so far...
I know that the total number of choices you can draw is 15, because (6 choose 2) equals 15.
A: {one of the balls is yellow} would equal 5/15.
B: {at least one ball is red} equals 9/15.
C: {both balls are green} equals 3/15.
D: {both balls are the same color} equals 4/15.

To find P(A|B), I have been trying to use the equation P(A|B) = P(A and B) / P(B). When I work this out, I get P(A|B) = 1/3. This is incorrect.

I don't really know how to proceed from here.

 

[galactus]

The top 4 are correct. Good work.

If you are using \(\displaystyle P(A|B)=\frac{P(A\text{and}B)}{P(B)}\)

Then, P(A and B)=prob. of one is yellow and at least one is red.

If one is yellow, then only one can be red, the remaining one.

So, what is the prob. one is yellow and one is red. 2/15

P(B)=prob. at least one is red. We have that as 3/5.

(2/15)/(3/5)=2/9.

P(D|B')=prob. both are same color given none are red.

P(D|B')=P(D and B')/P(B')=[(4/15)(2/5)]/(2/5)=4/15

P(D|C')=prob both are same color given none are green

P(D|C')=P(D and C')/P(C')

P(D and C')=prob. both are same and not green. They can only be both red then. Because there is only one yellow.

The prob. they are both red is 1/15

prob. not green is they both have to be red, or one yellow and one red. 1/5

(1/15)/(1/5)=1/3

Hope I done that right. I have been busy at the same time. Hope I wasn't too distracted.