chap2-3slidespt5

chap2-3slidespt5 - Equiprobable Probability Spaces...

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Unformatted text preview: Equiprobable Probability Spaces Probability function Recall... Sample space: S = { s 1 ,s 2 ,...,s N } . Probability masses: P ( { s i } ) = p i = 1 N for all i = 1 ,...,N For any event A ∈ F : P ( A ) = N ( A ) N where N ( A ) = number of elements in A . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 136 Equiprobable Probability Spaces Example Experiment: Roll four fair dice. Sample space: { 1111 , 1112 ,..., 2111 ,..., 6666 } Events algebra: F = P S Equiprobable space: P ( A ) = N ( A ) /N , ∀ A ∈ F . N = 6 4 . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 137 Equiprobable Probability Spaces Example What is the probability of no 6 when rolling four fair dice? Event A = { 1111 , 1112 ,..., 1115 , 2111 ,..., 5555 } . N ( A ) = 5 4 Probability: P ( A ) = 5 6 4 . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 138 Equiprobable Probability Spaces Example What is the probability of at least one 6 when rolling four fair dice? A : “at least one 6” A c : “no 6” ⇒ P ( A c ) = ( 5 6 ) 4 . P ( A ) = 1- P ( A c ) = 1- ( 5 6 ) 4 . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 139 Equiprobable Probability Spaces Example What is the probability of exactly one 6 when rolling four fair dice? Event A = { 6111 , 6112 ,..., 6115 , 1611 ,..., 5556 } . N ( A ) : ( 4 1 ) Ways to choose the position of 6 × 5 3 Ways to fill the other positions Probability: P ( A ) = ( 4 1 ) 5 3 6 4 . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 140 Equiprobable Probability Spaces Example What is the probability of two 6’s when rolling four fair dice? Event A = { 6611 , 6612 ,..., 6655 , 1661 ,..., 5566 } . N ( A ) : ( 4 2 ) Ways to choose the positions of 6’s × 5 2 Ways to fill the other positions Probability: P ( A ) = ( 4 2 ) 5 2 6 4 . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 141 Equiprobable Probability Spaces Example The Birthday problem : What is the probability that at least two students in a class of size n have the same birthday? Experiment: Ask all n students for their birthday Days ⇔ a number ∈ A = { 1 ,..., 365 } . Sample space: All n-tuples in A n Assumption: Equiprobable space. N = 365 n ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 142 Equiprobable Probability Spaces Example A : “At least two students have the same birthday” A c : “All students have different birthdays” N ( A c ) = P (365 ,n ) (assumption: n ≤ 365 ). Probability: P ( A ) = 1- P ( A c ) = 1- P (365 ,n ) 365 n = 1- 365!...
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This note was uploaded on 02/12/2012 for the course ECSE 305 taught by Professor Champagne during the Spring '09 term at McGill.

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chap2-3slidespt5 - Equiprobable Probability Spaces...

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