What's the difference between P(B given A) and P(A and B)

I'm a little desperate for help here, In my elementary stats class we've been working on this lab for quite a while - it goes like this.

We have picked out 40 m&ms randomly and what we got are these data values:
Blue 6 Brown 4 Green 6 Orange 15 Red 2 Yellow 7

Now, we have 7 probabilities we need to find, for both with replacement and without replacement, if we picked two m&ms out.

a. P(G1 and G2)
b. P(Y2 | G1)
c. P((G1 and Y2) Or (Y1 and G2))
d. P(Y2)
e. P((G1 and Y2) and (Y1 and G2))
f. P(no yellows on either draw)
g. P(doubles)
h. P(no doubles)

So I think I get a-c, but even then I'm not sure. When I get to d, then I get really confused, asking myself, uhh what's the difference between and and given?

For example, for d, the way I figured to do it is either 1. P((Y2|Y1 is yellow) or (Y2|1st choice is not yellow)) or 2. P((Y2 and Y1) or (Y2 and 1st choice is not yellow))

But I can't figure out what's the difference, and i'm also not sure whether the outer function is or or and? Can someone help me out here?

[QUOTE=mynameistruth;445902
For example, for d, the way I figured to do it is either 1. P((Y2|Y1 is yellow) or (Y2|1st choice is not yellow)) or 2. P((Y2 and Y1) or (Y2 and 1st choice is not yellow))

But I can't figure out what's the difference, and i'm also not sure whether the outer function is or or and? Can someone help me out here?[/QUOTE]

Suppose there is no difference? Write out how you will compute both, they may just give the same result.

CB

but the multiplication rule for given is P(A and B) / P(B)...but if it's and, it would be simply P(A) * P(B) or have I also gotten that wrong?

I drew a tree diagram, but making comparisons with the one in my book, it just gets me more confused!

In another situation, I can understand that given makes the sample size smaller, but in this case, I can't even tell if it's independent or not! I mean with replacement for sure, but without? We are treating it as if it is independent, so I guess it must be, but what I'm saying is I can't logically understand the difference between and and given in this case.

Probability that the m&m will be chosen will be yellow given the first one is red, and probability that the 2 m&ms chosen will be yellow and red - what's the big difference? Is it just that given is more specific? But there is no shrinking of sample size, as the common definition (in my book) states, or am I mistaken?

I'm not necessarily looking for the right answer here, but to understand this problem..