Elements of Statistical Methods Probability (Ch 3) - Statistics

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Conditional Probability: Formal DefinitionThe previous example with equally likely outcomes can be rewritten asP(A|B) =#(A ∩ B)#(S ∩ B)=#(A ∩ B)/#(S)#(B)/#(S)which motivates the following definition in the general case,not restricted to equally likely outcomes=P(A ∩ B)P(B)Definition: When A and B are any events with P(B) > 0, then we define theconditional probability of A given B byP(A|B) =P(A ∩ B)P(B)This can be converted to the multiplication or product ruleP(A ∩ B) = P(A|B)P(B) and P(A ∩ B) = P(B ∩ A) = P(B|A)P(A)provided that in the latter case we have P(A) > 0.23
Two Headed/Tailed CoinsAsk Marilyn: Suppose that we have three coins, one with two heads on it (HH),one with two tails on it (TT) and a fair coin with head and tail (HT).One of the coins is selected at random and flipped. Suppose the face up is Heads.What is the chance that the other side is Heads as well?We could reason as follows:1. Given the provided information, it can’t be the TT coin. It must be HH or HT.2. If HH was selected face down is Heads, if HT then face down is Tails3. Thus the chance of having Heads as face down is 1/2. Or is it?24
Page 1 and 2: Elements of Statistical MethodsProb
Page 3: Subjectivist Interpretation of Prob
Page 7 and 8: Collection of Events: Required Prop
Page 9 and 10: The Probability Measure PA probabil
Page 11 and 12: Countable Additivity =⇒ Finite Ad
Page 13 and 14: P(A ∪ B) = P(A) + P(B) − P(A
Page 15 and 16: Equally Likely OutcomesMany useful
Page 17 and 18: Combinatorial Counts40 patients in
Page 19 and 20: Rolling 5 Fair Dice (continued)c) W
Page 21 and 22: Matching or Adjacent BirthdaysWhat
Page 23: Conditional ProbabilityS● ● ●
Page 27 and 28: Applying the Multiplication RuleP({
Page 29 and 30: P(D|T + )?Typically something is kn
Page 31 and 32: IndependenceThe concept of independ
Page 33 and 34: Implied IndependenceIf A and B are
Page 35 and 36: Postulated IndependenceIn practical
Page 37 and 38: Mutual Independence of a Collection
Page 39 and 40: Random Variables for Two Coin Tosse
Page 41 and 42: Induced Events and ProbabilitiesSup
Page 43 and 44: CDF for Single Coin Toss or Coin Sp
Page 45 and 46: 2 Fair Coin TossesFor two fair coin
Page 47 and 48: General CDF Properties1. 0 ≤ F(y)
Page 49: Independent Random VariablesA colle

Elements of Statistical Methods Probability (Ch 3) - Statistics

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