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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 ntuples 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.
 Spring '09
 Champagne

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