1 / 11

Introduction

Introduction. Most environmental processes random Causes of randomness? Probability Theory: Framework for organizing information about randomness Outline: Motivating question Set notation Counting Conditional Probability Answer Question. Detecting a Leak.

chad
Download Presentation

Introduction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Introduction • Most environmental processes random • Causes of randomness? • Probability Theory: Framework for organizing information about randomness • Outline: • Motivating question • Set notation • Counting • Conditional Probability • Answer Question

  2. Detecting a Leak • Some underground storage containers leak. • Leak detection is not perfect (false positives, false negatives). • Test shows a leak has occurred. What is probability that a leak has actually occurred? • Test shows no leak. What is probability of leak?

  3. Data or Evidence • Simple summary statistics, based on laboratory experiments: • Pr(Leak occurs) = .10 • Pr(Leak detected given leak occurs) = .95 • Pr(leak detected given no leak occurs) = .10 (False +) • These imply: Pr(no leak)=.90, Pr(no leak detected given leak occurs)=.05 (False -), Pr(no leak detected given no leak occurs)=.90

  4. Sample Space and Events • Sample space: the set of all possible outcomes that might be observed. • Gender of first-born lion cub. • Selenium concentration in SB water supply. • Event: Some subset of the sample space. • “F” = event that first-born cub is female. • “B” = event that selenium concentration below EPA standard (.05 mg/L). • We usually define events in accordance with our ultimate question, then assign probabilities to events.

  5. Set Operations • Draw Set Diagram. • Union: A  B. The set of elements belonging to A or B or both. “Or” • Intersection: A  B. The set of elements belong to A and B. “And” • A  B, A  B, Ac  Bc

  6. Gender and Bicycles • Gender, Transportation example: • F=event that person is female • B=event that person rides bike to school. • Count # in F, B, F  B, F  B, N • Set Diagram: • P(F) = NF/N • P(B) = NB/N, • P(FB)=NFB/N

  7. Conditional Probability • Outcome of one random experiment may provide information about outcome of another random experiment • Green Party, Electric Vehicle. • Conditional Probability of event A given that event B has occurred: • P(A|B)=P(A  B)/P(B) … (see why on graph) • Select student at random. Given female, what is probability ride bicycle?: • P(B|F)?

  8. Bayes Theorem • Useful theorem for conditional probability. • Events E1,…, En are mutually exclusive and exhaustive. Then for any event A:

  9. Special Topic: Counting(Sewage Treatment Example) • Multiplication Principle: n1*n2 • 2 step sewage treatment: (3 physical*2 chemical=6) • Permutations of n things: n! • 5 possible biological treatments: (5!=120). • Permutations of n things r at a time: nPr=n!/(n-r)! (order matters) • 3 treatments from total of 5: (5!/2!=60). • Combinations of things r at a time: n!/(r!(n-r)!) (only list matters) • List of 3 from total of 5: (5!/(3!2!)=10).

  10. Solution to Leak Detection • Sample space: {detect, no detect}, {leak occurs, no leak occurs} • Events: • L=event that leak occurs • D=event that leak is detected • We want P(L|D)

  11. What we know • P(L)=.10, P(Lc)=.90 • P(D|L)=.95, P(Dc|L)=.05 • P(D|Lc)=.10, P(Dc|Lc)=.90 • Use Bayes Rule:

More Related