1 Probability Scott Matthews Courses: / / Lecture /17/2005

and Admin Issues  PS 4, Project 1 due Wednesday  Office hours tomorrow  Final Project Ideas (Brad& Don from FMS)  Lecture

and Probability  Only reviewing the more advanced concepts (and what we’ll need in course)  Basic concepts: between 0 and 1, additive, total probability must be 1, Venn diagrams, etc.

and Conditional Probability  “Probability (P) that A occurs conditional on B occurring”  Also referred to as “P of A given B”  Joint Probability: P(A and B) Cool MS equation

5 HIV Test Example: Table of Actual Condition, Test Results HIV-HIV+Total Test Result Positive Negative Total991100

and Conditional Probabilities  False Positive Test: P(HIV-|pos)  = 3.96/4.86 =.815  False Negative): P(HIV+|neg)  = 0.1/95.14 =.001

and Total Probability  Probability of an event occuring alone is combination of all possible joint outcomes with another event  Given n mutually exclusive events (A 1..A n ) whose probabilities sum to 1:

and Total Probability zSuppose the lightbulbs you can buy at store are manufactured by three factories. What is the total probability that a lightbulb sold at the store is defective? yFactory One produces 60% of the light bulbs sold yFactory Two produces 30% of the light bulbs sold yDefective bulb probabilities: 0.01 for Factory One, 0.02 for products of Factory Two, and 0.05 for products of Factory Three.

and Answer  P(defect) = P(defect|factory1)*P(factory1) + P(Defect|factory 2)*P(factory 2) +P(defect|factory 3)*P(factory 3)  =0.01* * *0.1 = =.017

and Bayes’ Theorem  “Opposite” of old conditional equation is:  But P(A and B) must equal P(B and A)..  So P(B|A)*P(A) = P(A|B)*P(B), thus  Using total probability..  Way of finding P(B|A) knowing only P(A|B)

and Bayes Example z+ event that drug test is positive for person z- event that drug test is negative for person zA event that person tested uses drug tested for  Assume P[A] .1, P[  |A] .98, P[  |Abar] .1  Bayes Theorem:

and Discrete Distributions  Values can only take on a set of countable values  Probability mass function (pmf) is map of probabilities of each possible outcome  Aka a histogram.  Cumulative distribution function (cdf) is P(X <= x)

and Discrete Dist’ns (cont.)  Should look familiar - recall lecture on risk profiles. Those were pmf’s, cdf’s.  basics12.htm

and Continuous distributions  Analogous to pmf/cdf for discrete case  Except pmf=> probability density function (pdf)

and Reading pdf/cdf graphs  What information can we see from just looking at a randomly selected pdf or cdf?

and Subjective Probabilities