slides4 - Lecture Stat 302 Introduction to Probability -...

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Unformatted text preview: Lecture Stat 302 Introduction to Probability - Slides 4 AD Jan. 2010 AD () Jan. 2010 1 / 24 Axioms of Probability Consider an experiment with sample space S . For each event E , we assume that a number P ( E ) , the probability of the event E , is de&ned and satis&es the following 3 axioms. Axiom 1 & P ( E ) & 1 Axiom 2 P ( S ) = 1 Axiom 3 . For any sequence of mutually exclusive events f E i g i 1 , i.e. E i \ E j = ? when i 6 = j , then P ( [ i = 1 E i ) = i = 1 P ( E i ) Direct consequences include P ( ? ) = 0 and for mutually exclusive events f E i g i 1 P ( [ n i = 1 E i ) = n i = 1 P ( E i ) . AD () Jan. 2010 2 / 24 Examples Example (coins): Assume both coins are unbiased; i.e. a head is as likely to appear as a tail, then P ( f H , H g ) = P ( f H , T g ) = P ( f T , H g ) = P ( f T , T g ) = 1 4 . Example : A die is rolled and we assume P ( f 1 g ) = P ( f 2 g ) = &&& = P ( f 6 g ) = 1 / 6 . Hence as a consequence from axiom 3, the probability of having an even or odd number is equal to P ( f 1 , 3 , 5 g ) = P ( f 1 g ) + P ( f 3 g ) + P ( f 5 g ) = 1 / 2 , P ( f 2 , 4 , 6 g ) = P ( f 2 g ) + P ( f 4 g ) + P ( f 6 g ) = 1 / 2 . AD () Jan. 2010 3 / 24 Properties Proposition : P ( E c ) = 1 & P ( E ) . We have S = E [ E c and E \ E c = ? so P ( S ) = 1 | {z } axiom 2 = P ( E [ E c ) = P ( E ) + P ( E c ) | {z } axiom 3 . Proposition : If E F then P ( E ) P ( F ) . We have F = E [ ( E c \ F ) and E \ (( E c \ F )) = ? so P ( F ) = P ( E ) + P ( E c \ F ) | {z } 0 by axiom 1 P ( E ) . Proposition : We have P ( E [ F ) = P ( E ) + P ( F ) & P ( E \ F ) . We have E [ F = E [ ( E c \ F ) and E \ (( E c \ F )) = ? so P ( E [ F ) = P ( E [ ( E c \ F )) = P ( E ) + P ( E c \ F ) but F = ( E c \ F ) [ ( E \ F ) with ( E c \ F ) \ ( E \ F ) = ? so P ( F ) = P ( E c \ F ) + P ( E \ F ) ) P ( E c \ F ) = P ( F ) & P ( E \ F ) . AD () Jan. 2010 4 / 24 Example You are in a restaurant and ordering 2 dishes. With proba 0.6, you will like the &rst dish; with proba 0.4, you will like the second dish. With proba 0.3, you will like both of them. What is the proba. you will like neither dish? Let A i the event: "You like dish i . Then the proba you like at least one is P ( A 1 [ A 2 ) = P ( A 1 ) + P ( A 2 ) & P ( A 1 \ A 2 ) = . 6 + . 4 & . 3 = . 7 . The event that you like neither dish is the complement of liking at least one, so P ( " you will like neither dish ) = P (( A 1 [ A 2 ) c ) = 1 & P ( A 1 [ A 2 ) = . 3 AD () Jan. 2010 5 / 24 Example A die is thrown twice and the number on each throw is recorded. Assuming the dice is fair, what is the probability of obtaining at least one 6? There are clearly 6 possible outcomes for the &rst throw and 6 for the second throw. By the counting principle, there are 36 possible outcomes for the two throws. Let A i the event I have obtained a 6 for throw i . The probability we are interested in is P ( A 1 [ A 2 ) = P ( A 1 ) + P ( A 2 ) & P ( A 1 \ A 2 ) = 1 6 + 1 6 & 1 36 = 11 36 ....
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slides4 - Lecture Stat 302 Introduction to Probability -...

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