Our rst theorem the probability of nota 1 probabilitya

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Unformatted text preview: e space •  We need a way of talking about “The outcome is one of a parAcular class.” •  Combining events. –  A or B is union –  A and B is intersecAon –  Not(A) is the complement Examples •  Everyone has systolic blood pressure above 200 mm mercury. •  The first person has systolic blood pressure of 120 mm mercury. •  At least two people have systolic blood pressure below 100 mm mercury. Combining events •  •  •  •  •  •  •  A or B = B or A A and B = B and A (A and B) and C = A and (B and C) (A or B) or C = A or (B or C) A and (B or C) = (A and B) or (A and C) Not(A and B) = Not(A) or Not(B) Not(A or B) = Not(A) and Not(B) The Axioms of Probability •  ProbabiliAes between 0 and 1 inclusive. –  The point: not infinite. •  The sample space has probability 1 –  Has to be something… •  P(DISJOINT union) = sum of probabiliAes –  (Disjoint means no overlap) –  This is where everything really starts! –  Infinite sums or finite sums Probability is like “weight” •  The weight of everything is 1 •  The weight of two “disjoint” pieces is the sum of the weights. Our first theorem •  The probability of not(A) = 1 - probability(A) •  Proof: –  P(A or not(A)) •  It’s P(the sample space)… so it’s 1 •  It’s P(A) + P(not(A)) –  Why? Because A and not(A) are disjoint. •  So, 1 = P(A) + P(not(A)) •  Which axioms did we use? A corollary •  The...
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