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Pg 13 - 6 Conditional Probability Let A and B be any events...

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Unformatted text preview: 6 Conditional Probability - Let A and B be any events in 8'. Suppose A depends on B, knowing B-adds information about A. The conditional probability of A given B is denoted P(A§BI) @4113) A 134(25):: H18 Tim ’55:) 0 Example: Suppose that among people buying a digital camera 60% purchase an op- tional memory card, 40% purchase an optional battery, and 30% purchase both the optional memory card and battery. ( a) Given a person purchases the battery, what is the probability the person also purchases the memory card? . ‘ -~_.. fin , 1‘ : .._, F .1. .413 r. . 1. 1‘5: hi. f} F 2 1’1“ I? W“ 39:: :fré C5!“ rd 4;}? ‘fi‘ if 11-“ ‘ P(/V\{‘l Ll": ‘ “'6 w a}: g .. ‘ - 5L» .5 1i “5) PM?) «'7 75 (b) Given a person purchaSes the memory card, What is the probability the person no 0 Let A be a set which consists of k disjoint subsets A1, . . . , Ah such that A: {A1, . . ”Ak} AZ- (1 Aj = 9 when i aé j (pairwise disjoint), and Al U . U At: A (the A form a . complete partition of A). The probability of A is given by (N A; "i H ; “3-: if” -;f : - E , ; "V; P(A) : P(A1)P( (A|.A1)+ P(Ak)P( A|Ak) Z P(A.-)P(AIA.-) 3’” J which is called the law of total probability for an event A. Here, the probability of the event A is not known directly. We use the conditional probability of A given AE- has occurred P(A}Ai), and the'individual. probability of Ai, P(A.;). - Example: The transmission of hits over a binary communication channel has the fol— lowing charanteristics: 0’s and 1’s have an equal chance of being sent. There is a 95% chance that a sent 0 will be received as 0. There is a 90% chance that a sent 1 Wiil be received as 1. <1}: ewe?“ is M9” ~ 5 ...
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