Lecture04 - Todays Schedule Probability notation laws...

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24 April 2003 Biostatistics 6650--L4 1 Today’s Schedule Probability notation laws example Screening test measures sensitivity/specificity positive(negative) predictive value ROC curves

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24 April 2003 Biostatistics 6650--L4 2 Probability “Nothing is impossible. Some things are just less likely than others.” Jonathan Winters
24 April 2003 Biostatistics 6650--L4 3 Probability What is a random process or variable? A process is random if individual outcomes are uncertain, but in an infinite sample there is a regular distribution of outcomes. Only one of 3 outcomes (A,B,C) can occur Theoretical frequency may be (.80,.19,.01) (.80,.10,.10) (.33,.33,.33), equally likely, basis of random sampling

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24 April 2003 Biostatistics 6650--L4 4 Probability What is probability ? Let the sample space be all possible outcomes blood types O, A, B, AB; coin flip: H, T; An event is any set of outcomes of interest let the event of interest be type A The probability of an event is the relative frequency of this set of outcomes over an indefinitely large number of trials (Rosner) type A is known to have a relative frequency (probability) of .27---in very large samples it is known that 27% of the population has blood type A.
24 April 2003 Biostatistics 6650--L4 5 Probability Notation/Law Let Pr(A) denote the prob. of event A occurring 0 <= Pr(A) <= 1 Pr(not A)= 1-Pr(A) If A and B are two mutually exclusive events (can’t both occur at the same time) then Pr(A or B)=Pr(A) + Pr(B). In general, Pr(A or B)=Pr(A) + Pr(B) - Pr(A and B) B A B A A Not A Not A and Not B Roll dice twice: A=6 on first, B=6 on 2nd Pr(A or B)=1/6 + 1/6 - (1/6)(1/6)=11/36

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24 April 2003 Biostatistics 6650--L4 6 Probability Notation/Law Set of events A 1 , A 2 ,…, A k are mutually exhaustive if at least one must occur. The sum of the probabilities for an exhaustive set must equal 1. 1 Pr( ) 1 k i i A = =
24 April 2003 Biostatistics 6650--L4 7 Probability Blood type example Type Probability O .49 A .27 B .20 AB ? Are blood types mutually exclusive? Pr(type AB) = Pr(type A or type B)= Pr(not type O)=

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24 April 2003 Biostatistics 6650--L4 8 Probability Notation/Law A and B are independent if Pr(A and B)=Pr(A) * Pr(B) A and B are dependent if Pr(A and B) Pr(A)*Pr(B) If A and B are independent , then Pr(A) is not influenced by whether or not B has occurred. This leads into conditional probability Two independent coin flips: Pr(H,H)=?(.5)*(.5)=.25, yes 4 outcomes(t/t, t/h, h/t, h/h) only one of which is h/h. Draw 2 cards from a deck of 52. What is Pr(K,K). With replacement: Pr (K,K)=(4/52)(4/52)=.0059-indep. Without replacement: Pr (K,K)=(4/52)(3/51)=.0045--dep.
24 April 2003 Biostatistics 6650--L4 9 Probability Conditional probability Let Pr(A|B) = conditional probability of A given that event B has occurred.

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