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Unformatted text preview: lustrate its use we consider:
Example A.5. Suppose we roll a four-sided die then ﬂip that number of coins.
What is the probability we will get exactly one Heads? Let B = we get exactly
one Heads, and Ai = an i appears on the ﬁrst roll. Clearly, P (Ai ) = 1/4 for
1 i 4. A little thought gives
P (B |A1 ) = 1/2, P (B |A2 ) = 2/4, P (B |A3 ) = 3/8, P (B |A4 ) = 4/16 so breaking things down according to which Ai occurs,
P (B ) = 4
i=1 = 1
4 P (B \ Ai ) = ✓ 4
2 4 8 16 P (Ai )P (B |Ai ) ◆ = 13
32 One can also ask the reverse question: if B occurs, what is the most likely
cause? By the deﬁnition of conditional probability and the multiplication rule,
P (Ai \ B )
P (Ai )P (B |Ai )
P (Ai |B ) = P4
j =1 P (Aj \ B )
j =1 P (Aj )P (B |Aj ) (A.5) This little monster is called Bayes’ formula, but it will not see much action
Last but far from least, two events A and B are said to be independent
if P (B |A) = P (B ). In words, knowing that A occurs does not change the
probability that B occurs....
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- Spring '10
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