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f2006lecNotesWeek08

f2006lecNotesWeek08 - 22 Probability(cont 22.1 Independent...

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Unformatted text preview: 22 Probability (cont.) 22.1 Independent events: P ( A ∩ B ) = P ( A ) P ( B ) — Epp § 6.9 • events A and B are independent iff P ( A ∩ B ) = P ( A ) P ( B ) • an example consider 3-child 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 — Epp § 6.8 • recall | A ∪ B | = | A | + | B | - | A ∩ B | show that this implies, for equally likely outcomes, P ( A ∪ B ) = P ( A ) + P ( B )- P ( A ∩ B ) • as it turns out, the result also applies to non-equally likely outcomes we will not prove this • 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 ∩ B ) = 0 • if P ( A ∩ B ) = 0 ( i.e. A ∩ B = ∅ ) then P ( A ∪ B ) = P ( A ) + P ( B ) A and B are mutually exclusive events • contrast independent and mutually exclusive make sure that they are clear mutually exclusive are very dependent 22.4 Conditional probability — Epp § 6.9 • general idea: if we have partial information about an experiment, it may modify our calculation of probability • 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 } • definition: P ( A | B ) = P ( A ∩ B ) P ( B ) how do we read this?...
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f2006lecNotesWeek08 - 22 Probability(cont 22.1 Independent...

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