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Unformatted text preview: probability of the empty set is zero! –  It’s impossible for nothing to happen •  Proof: The empty set and the sample space –  Together they make a disjoint union •  Whose probability is 1 –  Why? They are disjoint; their union is 1 •  What + 1 = 1? Another probability calculaAon •  P(A or B) = P(A) + P(B) – P(A and B) A •  Proof idea: –  The union is the three parts –  The sum has the middle part twice –  Use the “disjoint union” axiom. A and B B Special case: equally likely outcomes •  If there are n outcomes, then –  … each singleton set has probability 1/n. –  And any set has probability cardinality/n –  Proof: the singletons are disjoint and exhausAve –  and they all have equal probability –  x + x + … + x = 1, so x = 1/n And any set has probability cardinality/n •  So we are going to have to count cardinaliAes… CounAng I •  N ways to do the first thing •  and for each of those, •  M ways to do the second •  How many ways for the first and the second? N Ames M Choosing labeled balls from urns •  20 in the urn, and you intend to draw twice. •  How many different sequences? •  20 ways to choose the first. •  For each of those, •  19 ways to choose the second •  20 Ames 19 sequences. CounAng II •  •  •  •  •  N ways to choose first for each of them, M ways to choose second and for each of those, P ways to choose the third, … •  Number of ways to choose the sequence N Ames M Ames P Ames … CounAng III •  How many ways to order a set of size N? •  •  •  •  •  How many ways to choose the first? For each, how many for the second? For each, … For each, one way to choose the last. •  N (N- 1)(N- 2) … 1 •  If you have five playing cards, how many ways are there to arrange them? •  5 Ames 4 Ames 3 Ames 2 Ames 1 •  5! = 120 CounAng IV •  What about SETS, not SEQUENCES? •  How many ways to choose a set of size k from a set of size n? •  Trick: every set may be made into a unique sequences in k! ways. There are n(n- 1)(n- 2)….(n- k- 1) sequences. Set, not sequence, of size k, from n •  Trick: every set may be made into a unique sequences in k! ways. There are n(n- 1)(n- 2)….(n- k- 1) sequences. n(n- 1)(n- 2)…(n- k- 1)/k! = n!/(n- k)!k! How many ...
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This document was uploaded on 03/02/2014 for the course STATS 4150 at Barnard College.

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