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Unformatted text preview: ch side of (A.3) by P (A) gives us
the multiplication rule:
P (A \ B ) = P (A)P (B |A) (A.4) Intuitively, we think of things occurring in two stages. First we see if A occurs,
then we see what the probability B occurs given that A did. In many cases
these two stages are visible in the problem. 209 A.1. PROBABILITIES, INDEPENDENCE Example A.4. Suppose we draw two balls without replacement from an urn
with 6 blue balls and 4 red balls. What is the probability we will get two blue
balls? Let A = blue on the ﬁrst draw, and B = blue on the second draw.
Clearly, P (A) = 6/10. After A occurs, the urn has 5 blue balls and 4 red balls,
so P (B |A) = 5/9 and it follows from (A.4) that
P (A \ B ) = P (A)P (B |A) = 65
10 9 To see that this is the right answer notice that if we draw two balls without
replacement and keep track of the order of the draws, then there are 10 · 9
outcomes, while 6 · 5 of these result in two blue balls being drawn.
The multiplication rule is useful in solving a variety of problems. To il...
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