Lec15 Conditional Probability.pdf

# Lec15 Conditional Probability.pdf - Lec15 Conditional...

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Lecture15- MACM201-… Lec15 Conditional Probability Wednesday, February 22,2017 8:44 AM

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࠵? ( ࠵? | ࠵? ) = 0 ( p∩q ) 0 ( q ) ⎯⎯⎯⎯⎯⎯.….….….….….….….….… (1) ࠵? ( ࠵?|࠵? ) = ࠵? ( ࠵? ∩ ࠵? ) ࠵? ( ࠵? ) ⎯⎯⎯⎯⎯⎯⎯⎯⎯ ࠵? ( ࠵? ∩ ࠵? ) = ࠵? ( ࠵? ) W ࠵? ( ࠵? | ࠵? ) .….….….…... (2) Substitute (2) in (1) ࠵? ( ࠵? | ࠵? ) = ࠵? ( ࠵? ) W ࠵? ( ࠵? | ࠵? ) ࠵? ( ࠵? ) ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

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Let ࠵? be the event thata studentis male Let ࠵? be the event thata studentfailed calculus We wantto find ࠵? ( ࠵? ∩ ࠵? ) ࠵? ( ࠵? ) = 0.5 ࠵? ( ࠵? | ࠵? ) = 0.05 ࠵? ( ࠵? | ࠵? ) = ࠵? ( ࠵? ∩ ࠵? ) ࠵? ( ࠵? ) ⎯⎯⎯⎯⎯⎯⎯⎯⎯ ࠵? ( ࠵? ∩ ࠵? ) = ࠵? ( ࠵? | ࠵? ) ×࠵? ( ࠵? ) = 0.05×0.5 = 0.025

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Proof : Suppose ࠵? and ࠵? are independent. ࠵? ( ࠵? | ࠵? ) = ࠵? ( ࠵? ) . But ࠵? ( ࠵? | ࠵? ) = 0 ( 1∩2 ) 0 ( 2 ) ⎯⎯⎯⎯⎯⎯ .….….….….….….….…..(1) ∴ ࠵? ( ࠵? | ࠵? ) ×࠵? ( ࠵? ) = ࠵? ( ࠵? ∩ ࠵? ) .….….…. (2) Substitute(1) in (2) ࠵? ( ࠵? ∩ ࠵? ) = ࠵? ( ࠵? ) ×࠵? ( ࠵? ) Conversely, assume ࠵? ( ࠵? ∩ ࠵?

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