75 definition two events a and b are independent if p

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DEFINITION : Two events A and B are independent if P A B P A P B .If A and B are not independent then we say they are dependent . Unlike using conditional probabilities, the previous definition of independence allows the case where P A or P B is zero. In fact, it follows from this definition that any event with zero probability is independent of any other event . (The argument is simple: A B A so if P A 0 then P A B 0, and so the definition of independence is satisfied regardless of what P B is.) 76
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EXAMPLE : Draw a card from a standard 52 card deck. A card is a queen B card is a diamond A B card is queen of diamonds Now P A 4/52 1/13, P B 13/52 1/4, and P A B 1/52. So A and B are independent events. EXAMPLE : In the Monty Hall problem, A and B are dependent if p q . 77
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THEOREM :If A and B are independent events then so are (i) A and B c , (ii) A c and B , and (iii) A c and B c . Proof: To prove (i) we need to show P A B c P A P B c .But P A B c P A B P A P A B P A P A P B (by independence of A and B ) P A  1 P B  P A P B c . Parts (ii) and (iii) now follow by applying (i) and the symmetry of the definition of independence. 78
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Independence can be defined for a collection of more than two events. Let A 1 , A 2 , A 3 ,... be a countable collection of events (including a finite collection). This collection is said to be mutually independent if for every subset of events A i 1 , A i 2 ,..., A i k , P A i 1 A i 2 A i k P A i 1 P A i 2  P A i k This definition implies that the events are pairwise independent, but also much more. 79
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For three events A , B , and C , mutual independence requires P A B P A P B P A C P A P C P B C P B P C P A B C P A P B P C Mutual independence is handy for simplifying probability calculations. Typically, one assumes the events in question are independent, and proceed from there. (Casella and Berger has examples where the goal is to show whether mutual independence holds.) 80
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EXAMPLE : Flip a fair coin n times. The probability of a head on each flip is 1/2. Assume that the flips are mutually independent. Formally, let A j flip j produces a head . Then P n heads in a row P A 1 A 2 A n P A 1 P A 2  P A n (by assumed independence) 1 2 1 2 1 2 1 2 n Question: If I flip a fair coin ten times and get all heads, what is the probability that the 11 th flip is a head?
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75 DEFINITION Two events A and B are independent if P A B P...

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