Lecture2 - Using the coin tossing experiment, and from...

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1 Lecture 2 ECE 514 ¾ Using the coin tossing experiment, and from experience we know that if we keep tossing a coin, eventually, a head must show up, i.e., But and using axiom 4 (generalization of axiom 3) of probability . 1 ) ( = A P = = 1 , n n A A ). ( ) ( 1 1 = = = = n n n n A P A P A P ¾ In a fair coin since only one in 2 n outcomes is in favor of A n , we have Agreeing with axiom 2 of probability ¾ In summary, we have a nonempty set Ω of outcomes ( elementary events), a σ -field F of subsets of Ω (combining to form events), and a probability measure P on the sets in F subject to the four axioms forming a triplet ( Ω , F, P ) i.e. a probability model . ¾ The probability of more complicated events must follow from this framework by deduction. , 1 2 1 ) ( and 2 1 ) ( 1 1 = = = = = n n n n n n A P A P Conditional Probability and Independence ¾ In N independent trials, suppose N A , N B , N AB denote the number of times events A , B and AB occur respectively. ¾ Using the frequency interpretation of probability, for large N ¾ In N A occurrences of A, only N AB of them are also found in the N B occurrences of B, giving the ratio . ) ( , ) ( , ) ( N N AB P N N B P N N A P AB B A ) ( ) ( / / B P AB P N N N N N N B AB B AB = = as a measure of “the event A given that B has already occurred”. We denote this conditional probability by P ( A | B ) = Probability of “the event A given that B has occurred”. ¾ We define provided ¾ For a conditional probability to be a probability, it has to satisfy all probability axioms discussed earlier. , ) ( ) ( ) | ( B P AB P B A P = . 0 ) ( B P We have (i) (ii) since Ω B = B . (iii) Suppose Then But hence satisfying all probability axioms , making a cond. Prob. a legitimate probability measure. , 0 0 ) ( 0 ) ( ) | ( > = B P AB P B A P . ) ( ) ( ) ( ) ) (( ) | ( B P CB AB P B P B C A P B C A P = = , 1 ) ( ) ( ) ( ) ( ) | ( = = Ω = Ω B P B P B P B P B P ), | ( ) | ( ) ( ) ( ) ( ) ( ) | ( B C P B A P B P CB P B P AB P B C A P + = + = . 0 = C A AB BC , ∩= φ ). ( ) ( ) ( CB P AB P CB AB P + =
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2 Properties of Conditional Probability: a. If and since if then occurrence of B implies automatic occurrence of the event A . As an example, but in a dice tossing experiment. Then and b. If and , , B AB A B = 1 ) ( ) ( ) ( ) ( ) | ( = = = B P B P B P AB P B A P , A B ). ( ) ( ) ( ) ( ) ( ) | ( A P B P A P B P AB P B A P > = = , , A AB B A = , A B . 1 ) | ( = B A P {outcome is even}, ={outcome is 2}, A B = ¾ In a dice experiment, so that ¾ The statement that B has occurred (outcome is even) makes “outcome is 2” more likely than without that information ¾ A conditional probability may be used to simplify the probability of a complicated event (in terms of “simpler” related events) ¾ Let be pair wise disjoint and their union is Ω . Thus and Thus . B A . 1 Ω = = n i i A n A A A , , , 2 1 " , φ = j i A A . ) ( 2 1 2 1 n n BA BA BA A A A B B = = " " {outcome is 2}, ={outcome is even}, A B = ¾ But so that the union expression above yields ¾ Along with the notion of conditional probability, we introduce the notion of “independence” of events
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Lecture2 - Using the coin tossing experiment, and from...

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