- #1

TheMathNoob

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## Homework Statement

Selecting Two Balls. Suppose that two balls are to be selected at random, without replacement, from a box containing r red balls and b blue balls. We shall determine the probability p that the ﬁrst ball will be red and the second ball will be blue

I am confusing Pr(A|B) and Pr(A∩B). In some cases they might look the same to me, so how can I recognize the difference between them.

## Homework Equations

Pr(A∩B)=Pr(A) Pr(B|A)

Pr(A∩B)=Pr(B) Pr(A|B).

## The Attempt at a Solution

This is an example from the book, so the solution is already there.

Let A be the event that the ﬁrst ball is red,and let B be the event that the second ball is blue. Obviously

, Pr(A)=r/(r +b)

Furthermore, if the event A has occurred, then one red ball has been removed from the box on the ﬁrst draw. Therefore, the probability of obtaining a blue ball on the second draw will be

Pr(B|A)= b/( r +b−1)

It follows that

Pr(A∩B)= r/( r +b)* (b/( r +b−1))

So the line " Furthermore, if the event A has occurred, then one red ball has been removed from the box on the ﬁrst draw. Therefore, the probability of obtaining a blue ball on the second draw will be " looks the same to me as Pr(AnB) because the two events have to have occurred.

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