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Unformatted text preview: 22 Probability (cont.) 22.1 Independent events: P ( A âˆ© B ) = P ( A ) P ( B ) â€” Epp Â§ 6.9 â€¢ events A and B are independent iff P ( A âˆ© B ) = P ( A ) P ( B ) â€¢ an example consider 3child families assign equal probabilities to GGG, GGB, ... , BBB (all 1 8 ) let A be: 1st child is a girl let B be: even number of girls let C be: 2 girls show that A and B are independent show that A and C are not independent 22.2 Inclusion/exclusion applied to probability â€” Epp Â§ 6.8 â€¢ recall  A âˆª B  =  A  +  B    A âˆ© B  show that this implies, for equally likely outcomes, P ( A âˆª B ) = P ( A ) + P ( B ) P ( A âˆ© B ) â€¢ as it turns out, the result also applies to nonequally likely outcomes we will not prove this â€¢ an example draw a card at random from an ordinary deck find probability of either a heart or an ace 22.3 Mutually exclusive events: P ( A âˆ© B ) = 0 â€¢ if P ( A âˆ© B ) = 0 ( i.e. A âˆ© B = âˆ… ) then P ( A âˆª B ) = P ( A ) + P ( B ) A and B are mutually exclusive events â€¢ contrast independent and mutually exclusive make sure that they are clear mutually exclusive are very dependent 22.4 Conditional probability â€” Epp Â§ 6.9 â€¢ general idea: if we have partial information about an experiment, it may modify our calculation of probability â€¢ e.g. families of 2 children use Venn diagrams with S = { B 1 B 2 , B 1 G 2 , G 1 B 2 , G 1 G 2 } assuming outcomes are equally likely, show P (2 boys) = 1 4 show that, given that there is at least one boy, P (2 boys) = 1 3 use S = { B 1 G 2 , G 1 B 2 , B 1 B 2 } similarly, given that the first child is a boy, P (2 boys) = 1 2 use S = { B 1 G 2 , B 1 B 2 } â€¢ definition: P ( A  B ) = P ( A âˆ© B ) P ( B ) how do we read this?...
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 Fall '06
 Carter
 Conditional Probability, ... ..., EPP

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