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

1. ## 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?

2. [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

3. 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..