If a b then x b x a a x b with the two sets

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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 X i=1 = 1 4 P (B \ Ai ) = ✓ 4 X i=1 123 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 = 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 here. 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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