Examples Involving Conditional Probability

8 prg1 41 prg1 4 prg1 4 72 math 30530

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Unformatted text preview: ance that I’ll go tomorrow I go to the gym today. Question: What’s the probability that I go 30 days from now? Answer: Let Gi = {I go to the gym on day i } (today is day 0) Pr(G0 ) = 1 Pr(G1 ) = .8 Math 30530 (Fall 2012) Conditional examples September 5, 2013 4/5 Am I at the gym? c Pr(G2 ) = Pr(G2 ∩ G1 ) + Pr(G2 ∩ G1 ) c c = Pr(G1 ) Pr(G2 |G1 ) + Pr(G1 ) Pr(G2 |G1 ) = .8 Pr(G1 ) + .4(1 − Pr(G1 )) = .4 Pr(G1 ) + .4 (= .72) Math 30530 (Fall 2012) Conditional examples September 5, 2013 5/5 Am I at the gym? c Pr(G2 ) = Pr(G2 ∩ G1 ) + Pr(G2 ∩ G1 ) c c = Pr(G1 ) Pr(G2 |G1 ) + Pr(G1 ) Pr(G2 |G1 ) = .8 Pr(G1 ) + .4(1 − Pr(G1 )) = .4 Pr(G1 ) + .4 (= .72) c c Pr(G3 ) = Pr(G2 ) Pr(G3 |G2 ) + Pr(G2 ) Pr(G3 |G2 ) = .8 Pr(G2 ) + .4(1 − Pr(G2 )) = .4 Pr(G2 ) + .4 (= .688) Math 30530 (Fall 2012) Conditional examples September 5, 2013 5/5 Am I at the gym? c Pr(G2 ) = Pr(G2 ∩ G1 ) + Pr(G2 ∩ G1 ) c c = Pr(G1 ) Pr(G2 |G1 ) + Pr(G1 ) Pr(G2 |G1 ) = .8 Pr(G1 ) + .4(1 − Pr(G1 )) = .4 Pr(G1 ) + .4 (= .72) c c Pr(G3 ) = Pr(G2 ) Pr(G3 |G2 ) + Pr(G2 ) Pr(G3 |G2 ) = .8 Pr(G2 ) + .4(1 − Pr(G2 )) = .4 Pr(G2 ) + .4 (= .688) In general, c c Pr(Gn ) = Pr(Gn−1 ) Pr(Gn |Gn−1 ) + Pr(Gn−1 ) Pr(Gn |Gn−1 ) = .8 Pr(Gn−1 ) + .4(1 − Pr(Gn−1 )) = .4 Pr(Gn−1 ) + .4 Math 30530 (Fall 2012) Conditional examples September 5, 2013 5/5 Am I at the gym? c Pr(G2 ) = Pr(G2 ∩ G1 ) + Pr(G2 ∩ G1 ) c c = Pr(G1 ) Pr(G2 |G1 ) + Pr(G1 ) Pr(G2 |G1 ) = .8 Pr(G1 ) + .4(1 − Pr(G1 )) = .4 Pr(G1 ) + .4 (= .72) c c Pr(G3 ) = Pr(G2 ) Pr(G3 |G2 ) + Pr(G2 ) Pr(G3 |G2 ) = .8 Pr(G2 ) + .4(1 − Pr(G2 )) = .4 Pr(G2 ) + .4 (= .688) In general, c c Pr(Gn ) = Pr(Gn−1 ) Pr(Gn |Gn−1 ) + Pr(Gn−1 ) Pr(Gn |Gn−1 ) = .8 Pr(Gn−1 ) + .4(1 − Pr(Gn−1 )) = .4 Pr(Gn−1 ) + .4 Pr(G30 ) ≈ 2 + 10−12 , and for n above about 10, Pr(Gn ) basically 3 indistinguishable from 2/3 Math 30530 (Fall 2012) Conditional examples September 5, 2013 5/5...
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This note was uploaded on 01/31/2014 for the course MATH 30530 taught by Professor Hind,r during the Fall '08 term at Notre Dame.

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