Independent events Two events are independent if knowing that one event is true or has happened does not change the probability of the other event. “male”

Slides:



Advertisements
Similar presentations
Multiplication Rule We now discuss the situation when both events A and B occur.
Advertisements

Multiplication Rule: Basics
Chapter 4 Probability: Probabilities of Compound Events
COUNTING AND PROBABILITY
Conditional Probability and Independence. Learning Targets 1. I can calculate conditional probability using a 2-way table. 2. I can determine whether.
7/20 The following table shows the number of people that like a particular fast food restaurant. 1)What is the probability that a person likes Wendy’s?
Objectives (BPS chapter 12) General rules of probability  Independence and the multiplication rule  The general addition rule  Conditional probability.
Objectives (BPS chapter 12)
Chapter 12: General Rules of Probability STAT 1450.
Chapter 3 Section 3.3 Basic Rules of Probability.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 6.1 Chapter Six Probability.
Statistics Lecture 6. Last day: Probability rules Today: Conditional probability Suggested problems: Chapter 2: 45, 47, 59, 63, 65.
Lecture 6. Bayes Rule David R. Merrell Intermediate Empirical Methods for Public Policy and Management.
Lesson #9 Rules of Probability. Venn Diagram S A.
Applying the ideas: Probability
Agresti/Franklin Statistics, 1 of 87 Chapter 5 Probability in Our Daily Lives Learn …. About probability – the way we quantify uncertainty How to measure.
Conditional Probability: What’s the Probability of A, Given B?
(Medical) Diagnostic Testing. The situation Patient presents with symptoms, and is suspected of having some disease. Patient either has the disease or.
Probability and inference General probability rules IPS chapter 4.5 © 2006 W.H. Freeman and Company.
5.3A Conditional Probability, General Multiplication Rule and Tree Diagrams AP Statistics.
Agresti/Franklin Statistics, 1 of 87 Chapter 5 Probability in Our Daily Lives Learn …. About probability – the way we quantify uncertainty How to measure.
Math The Multiplication Rule for P(A and B)
Probability and inference General probability rules IPS chapter 4.5 © 2006 W.H. Freeman and Company.
Topic 2 – Probability Basic probability Conditional probability and independence Bayes rule Basic reliability.
Probability and inference General probability rules IPS chapter 4.5 © 2006 W.H. Freeman and Company.
Probability Theory General Probability Rules. Objectives General probability rules  Independence and the multiplication rule  Applying the multiplication.
Bennie D Waller, Longwood University Probability.
CHAPTER 3 Probability Theory Basic Definitions and Properties Conditional Probability and Independence Bayes’ Formula Applications.
Probability. Statistical inference is based on a Mathematics branch called probability theory. If a procedure can result in n equally likely outcomes,
Warm-up – for my history buffs…  A general can plan a campaign to fight one major battle or three small battles. He believes that he has probability 0.6.
Basic Probability Rules Let’s Keep it Simple. A Probability Event An event is one possible outcome or a set of outcomes of a random phenomenon. For example,
Probability Definition: Probability: the chance an event will happen. # of ways a certain event can occur # of possible events Probability =  Probability.
1 Chapter 4, Part 1 Repeated Observations Independent Events The Multiplication Rule Conditional Probability.
Recap from last lesson Compliment Addition rule for probabilities
Reading The “Given That” Versus The “AND” Statement By Henry Mesa.
Conditional Probability and Independence. Learning Targets 1. I can use the multiplication rule for independent events to compute probabilities. 2. I.
Representing Data for Finding Probabilities There are 35 students 20 take math 25 take science 15 take both Venn Diagram Contingency table M^M.
Essential Statistics Chapter 111 General Rules of Probability.
Probability The Study of Randomness The language of probability Random is a description of a kind of order that emerges only in the long run even though.
The Study of Randomness
Section Conditional Probability Objectives: 1.Understand the meaning of conditional probability. 2.Learn the general Multiplication Rule:
Psychology 202a Advanced Psychological Statistics September 24, 2015.
In a random event, outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions. We define the.
Stat 1510: General Rules of Probability. Agenda 2  Independence and the Multiplication Rule  The General Addition Rule  Conditional Probability  The.
Stat 13, Thu 4/19/ Hand in HW2! 1. Resistance. 2. n-1 in sample sd formula, and parameters and statistics. 3. Probability basic terminology. 4. Probability.
Probability The Study of Randomness The language of probability Random is a description of a kind of order that emerges only in the long run even though.
Section 5.3 Independence and the Multiplication Rule.
CHAPTER 12 General Rules of Probability BPS - 5TH ED.CHAPTER 12 1.
Section 5.3: Independence and the Multiplication Rule Section 5.4: Conditional Probability and the General Multiplication Rule.
Probability The Study of Randomness The language of probability Random is a description of a kind of order that emerges only in the long run even though.
STAT 240 PROBLEM SOLVING SESSION #2. Conditional Probability.
6.2 – Probability Models It is often important and necessary to provide a mathematical description or model for randomness.
Probability What is the probability of rolling “snake eyes” in one roll? What is the probability of rolling “yahtzee” in one roll?
The Role of Probability Chapter 5. Objectives Understand probability as it pertains to statistical inference Understand the concepts “at random” and “equally.
Probability Class Homework Check Assignment: Chapter 7 – Exercise 7.20, 7.29, 7.47, 7.48, 7.53 and 7.57 Reading: Chapter 7 – p
The law of total probability and Bayes Formula. Law of Total Probability F1F1 F2F2 F3F3 E Suppose a sample space S is the union of n pairwise disjoint.
10. General rules of probability
9. Introducing probability
Warm-up – for my history buffs…
Aim: What is the multiplication rule?
Independent events Two events are independent if knowing that one event is true or has happened does not change the probability of the other event. “male”
The study of randomness
ASV Chapters 1 - Sample Spaces and Probabilities
The Study of Randomness
Independent and Dependent Events
Learn to let go. That is the key to happiness. ~Jack Kornfield
Dates of … Quiz 5.4 – 5.5 Chapter 5 homework quiz Chapter 5 test
Independent vs. Dependent events
6.2 Independence and the Multiplication Rule
Conditional Probability and the Multiplication Rule
Presentation transcript:

Independent events Two events are independent if knowing that one event is true or has happened does not change the probability of the other event. “male” and “get head when flipping a coin”  ?? “male” and “taller than 6 ft”  ?? “male” and “pregnant”  ?? This is completely different from the idea of events that are disjoint.

Sampling without replacement Pick one frog at random from your target population and don’t put it back. Then pick another frog, etc. Artificial pond with 10 male and 10 female frogs. P(1st frog is male) = ??. If 1st frog is male, P(2nd frog is male) ??.  Here, successive picks are ??. Survey of a whole county with thousands of frogs (half males, half females). P(1st frog is male) = ??. If 1st frog is male, P(2nd frog is male) ≈ ??.  Here, successive picks are “nearly” ??.

Conditional probability Conditional probabilities reflect how the probability of an event can be different if we know that some other event has occurred or is true. The conditional probability of event B, given event A is: (provided that P(A) ≠ 0) When two events A and B are independent, P(B | A) = P(B). No information is gained from the knowledge of event A.

Probabilities of hearing impairment and blue eyes among Dalmatian dogs. HI = Dalmatian is hearing impaired B = Dalmatian is blue eyed Neither HI nor B 0.66 B and not HI 0.06 HI and B 0.05 HI and not B 0.23 P(HI and B) = .05 P(HI) = ?? P(B) = ?? P(HI | B) = P(HI and B) / P(B) = ?? P(HI | B) ≠ P(HI ), therefore HI and B are ??.

Large-scale surveys indicate that 11% of the population smokes Large-scale surveys indicate that 11% of the population smokes. Medical researchers know that the probability that a smoker will get lung cancer is 0.34. The probability that a person will get lung cancer if the person doesn’t smoke is 0.03. The probability P(Lung cancer | Smoker) is A) 0.03 B) 0.0394 C) 0.34 D) 0.583 E) 0.97 Are “Smoker” and “Lung cancer” independent?? A) Yes B) No Why??

Non-smoker and no Lung cancer Large-scale surveys indicate that 11% of the population smokes. Medical researchers know that the probability that a smoker will get lung cancer is 0.34. The probability that a person will get lung cancer if the person doesn’t smoke is 0.03. Lung cancer Non-smoker and no Lung cancer Smoker 0.11 0.34 = P(Lung cancer | Smoker) is the fraction of smokers (the orange circle) that will get lung cancer (the blue dots that are within the orange circle). This is why by definition P(Lung cancer | Smoker) = P(Lung cancer and Smoker)/P(Smoker).

Multiplication rule General multiplication rule: The probability that any two events, A and B, both occur is: P(A and B) = P(A)P(B|A) Multiplication rule for independent events: If A and B are independent, then: P(A and B) = P(A)P(B)

Artificial pond with 10 male and 10 female frogs Successive captures are not independent. Probability of randomly capturing 2 male frogs in a row: P(male and male) =?? Blood donation center Unrelated visitors are independent. Probability that the next two unrelated visitors are both type O: P(O and O) = ??

Tree diagrams Tree diagrams are used to represent probabilities graphically and facilitate computations. Probabilities of skin cancer among men and women by body locations. Man Woman In a random individual with skin cancer: P(head) = ?? P(trunk) = ?? P(limbs) = ?? % in each group who are women: P(woman | head) = ?? P(woman | trunk) = ?? P(woman | limbs) = ?? 0.44 0.56 Head Trunk Limbs 0.15 0.44 0.63 0.37 Man Woman 0.41 0.20 0.80 Man Woman

Skin cancer example) P(H) = ??; P(H|M) = ?? Bayes’s Theorem: 𝑃 𝐴 𝑗 𝐵 = 𝑃 𝐵∩ 𝐴 𝑗 𝑃(𝐵) = 𝑃( 𝐴 𝑗 )𝑃(𝐵| 𝐴 𝑗 ) 𝑖=1 𝑘 𝑃( 𝐴 𝑖 )𝑃(𝐵| 𝐴 𝑖 ) where A1, A2, … , Ak nonzero mutually exclusive & exhaustive events, B any other event w/ P(B)≠0, P(B)≠1 but P(B) not given directly. P(Aj) prior probability; 𝑃( 𝐴 𝑗 |𝐵) posterior probability. Skin cancer example) P(H) = ??; P(H|M) = ?? Q) Mutually exclusive and exhaustive events that include H?? Q) Role of M?? How to compute P(M)?? P(M) = ?? P(H|M) = ??

Diagnosis tests Diagnosis sensitivity Disease prevalence No disease Individual Positive Negative True positive False negative False positive True negative If a person gets a positive test result, what is the probability that he/she actually has the disease? This is the positive predictive value: PPV = P(disease | positive test) Diagnosis specificity

HIV-AIDS: Suppose that about 1% of a large population has HIV-AIDS antibodies. The enzyme immuno-assay test has sensitivity .9985 and specificity .9940. Diagnosis sensitivity Disease prevalence Positive True positive .9985 HIV antibodies .01 .0015 Negative False negative Random adult taking the test Positive False positive .99 .006 No HIV antibodies Incidence of HIV-AIDS in the general population .994 Negative True negative Diagnosis specificity HIV-test performance If a person who takes this test gets a positive test result, what is the probability that he or she actually has HIV-AIDS, P(HIV-AIDS | positive test)?

PPV = P(disease | positive) Diagnosis sensitivity What’s the positive predictive value of the enzyme immuno-assay test for HIV-AIDS? PPV = P(disease | positive) Disease incidence .9985 Positive HIV antibodies .01 .0015 Negative False negative Random adult taking the test .0060 Positive False positive .99 No HIV antibodies Incidence of HIV-AIDS in general population .9940 Negative Diagnosis specificity HIV-test performance P(true positive) = P(disease and positive) = P(disease)P(positive | disease) = ?? P(false positive) = P(no disease and positive) = P(no disease)P(positive | no disease) =?? PPV = P(disease | positive) = P(disease and positive) / P(positive) = P(true positive) / P(all positives) =??

What is the rate of prostate cancer among mature men? P(cancer)= ?? How “sensitive” is the PSA test? P(+ | cancer)= ?? How “specific” is the PSA test? P(– | no cancer)= ?? If a man gets an elevated PSA test (+), what is the probability that he has prostate cancer? PPV = P(cancer | +)= ??