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# Bayesian Probablity

Bayesian probability is a different way of looking at the concept of probability. In Bayesian thinking, the probability of an event is not fixed, it is in fact just a quantification of your personal belief. Thus, it is only rational for you to 'update' and change what you believe to be the probability of an event whenever new evidence arrives. Bayesian probability largely relies on Baye's theorem, which can even be applied sequentially until a solution is arrived at. Here is the formula for the probability of event A occurring given event B has occurred: Here is an example of Bayesian updating with Baye's theorem in action. Consider a population with 1% of people have HIV. Take one person and you would assume that there is a 1% chance he has HIV. Now imagine there is an HIV test that is 99% accurate and this person tests positive. An incorrect updating of probability would say that this person has a 99% chance of having HIV. Instead it should look like this: What is the chance that someone tests positive given he has HIV? 99%.

What is the chance that someone has HIV? 1%

What is the probability that someone tests positive? Someone would test positive if they were in fact HIV positive and the test registered accurately (1% * 99%) or if they were in fact HIV negative and the test performed inaccurately (99% * 1%)

So you get: Bayesian updating shows that this person will only have HIV with probability 50%!

### Find the probability using Bayesian theorem

A certain disease occurs in 8% of the population. A test for the disease is fairly accurate: it misclassifies people with the disease as healthy 5% of the time and reports that a healthy person is diseased just 5% of the time. Suppose that a person tests positive for the disease. Compute the probability that the person does not

### Calculating Conditional Probability in Decision Making Case

1. A medical test for HIV infection has three possible test results: positive ( ), negative ( ), and inconclusive ( ). Given that a person is infected, the probabilities of these outcomes are: 0.85, 0.10, and 0.05, respectively. Given that a person is not infected, the probabilities of these outcomes are:

### Bayes Theorem for invasive cancer research

See attached file for clarity. Given: 8 cases of invasive cancer and a total years of employment of 145 teachers. The cancer institute statistcis suggest 4.2 cases of cancer could be expected to occur. Assumptions: 145 employees develop or do not develop cancer independently and chances of cancer for each employee is the

### Applied Statistics: Bayes Theorem for certain disease occurrence in 35% of population

A certain disease occurs in 35% of the population. A test for the disease is fairly accurate: it misclassifies people with the disease as healthy 6% of the time and reports that a healthy person is diseased just 7% of the time. Suppose that a person tests positive for the disease. Compute the probability that the person does ind

### Use of Bayes's rule: Use of testosterone in athletes

In a population of 1,000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive. Of the nonusers, suppose 9 would test positive. a) Given that the athlete is a user, find the probablility that a drug test would yield a positive result. b) Given that the athlete is a nonuser,

### A Series of Probability Questions

1. Jen will call Cathy on Saturday with a 60% probability. She will call Cathy on Sunday with an 80% probability. The probability that she will call on neither of the two days is 10%. What is the probability that she will call on Sunday if she calls on Saturday? 2. At a parking lot, there are 12 spaces arranged in a row. A ma

### Probability

1. P(A1) = .20, P(A2) = .40, and P(A2) = .40. P(B1|A2) = .25. P(B1|A2) = .05, and P(B1|A3) =.10. Use Bayes' theorem to determine P(A3 | B1). 2. Dr. Stallter has been teaching basic statistics for many years. She knows that 80 percent of the students will complete the assigned problems. She has also determined that among those

### Probability of selecting blood type A+; P(A) = 3/5; students with credit card, Bayes

1. The distribution of blood types for 100 Americans is listed in the table. If one donor is selected at random, find the probability of selecting with blood type A+. blood type O+ O- A+ A- B+ B- AB+ AB- number 37 6 34 6 10 2 4 1 A. 0.45 B. 0.34 C. 0.68 D. 0.4 The P(A) = 3/5. Find the odds in fav

### ELISA Test for AIDS and Bayes' Rule

The ELISA test for AIDS was used in the screening of blood donations in the 1990s. As with most medical diagnostic tests, the ELISA test is not infallible. If a person actually carries the AIDS virus, experts estimate that this test gives a positive result 97.7% of the time. If a person does not carry the AIDS virus, ELISA gives

### Statistics: Use Bayes' rule to find a table of all posterior probabilities for drug testing

Question 10) In the drug testing, assume there are three possible test results: positive, negative, and inconclusive. For a drug user, the probabilities of these outcomes are 0.65. 0.06, and 0.29. For a nonuser, they are 0.03, 0.72, and 0.25. Use Bayes' rule to find a table of all posterior probabilities. (The prior probability

### Bayes' Theorem: Uses for Business-related Questions

Why is using Bayes' theorem important to help answer business-related questions? What does this theorem allow you to do that traditional statistics do not? What are some prerequisites for using Bayesian statistics?

### Probability: Binomial Distribution and Bayes theorem

1. A Tamiami shearing machine is producing 10 percent defective pieces, which is abnormally high. The quality control engineer has been checking the output by almost continuous sampling since the abnormal condition began. What is the probability that in a sample of 10 pieces: a. Exactly 5 will be defective? b. 5 or more will b

### Why is using Bayes' theorem important

Why is using Bayes' theorem important to help answer business-related questions? What does this theorem allow you to do that traditional statistics do not? What are some prerequisites for using Bayesian statistics?

### Bay's Theorem

A special test is given to people complaining of sever headaches. If someone has a brain tumor, the test is positive with a probability of .8 (80%). If they have no tumor, the test is positive with a probability of .1 (10%). only 4% (.04) of those tested actually have brain tumors. What is the probability of someone who tested p

### Set of probability questions involving Venn diagrams, decision trees, and mutual exclusive events.

1. In each one of the following cases, indicate classic probability is used, empirical (relative occurrence frequency) or subjective. a. A basketball player fails 30 of 50 distances. The probability that he fails the following shot is .60. b. A committee of seven students forms to study the environmental questions. What is t

### Bayesian Statistics

A hospital receives two-fifths of its flu vaccine from Company A and the remainder from Company B. Each shipment contains a large number of vials of vaccine. From Company A, 3% of the vials are ineffective; from Company B, 2% are ineffective. A hospital tests n = 25 randomly selected vials from one shipment and finds that 2 are

### Bayes' Theorem - Sample Question

1.6-12. A test indicates the presence of a particular disease 90% of the time when the disease is present and the presence of the disease 2% of the time when the disease is not present. If 0.5% of the population has the disease, calculate the conditional probability that a person selected at random has the disease if the test

### Bayes' Theorem - DVDs & Disease

1.6-8. A store sells four brands of DVD players. The least expensive brand, B1, accounts for 40% of the sales. The other brands (in order of their price) have the following percentages of sales: B2, 30%; B3, 20%; and B4, 10%. The respective probabilities of needing repair during warranty are 0.10 for B1, 0.05 for B2, 0.03 for B

### Probability and the Bayes Theorem

1. The information below is the number of daily emergency service calls made by the volunteer ambulance service of Walterboro, South Carolina, for the last 50 days. To explain there were 22 days on which there were 2 emergency calls, and 9 days on which there were 3 emergency calls. Number of calls

### Using the Bayes Theorem

P(A?)=.20,P(A?)=.40,andP(A?)=.40.P(B?A?)=.25.P(B?A?)=.05, and P(B?A?)=.10. Use Bayes' theorem to determine P(A?B?). Dr. Stallter has been teaching basic statistics for many years. She knows that 80 percent of the students will complete the assigned problems. She has also determined that among those who do their assignments

### Bayes' Theorem and a Survey of Probability Concepts

A Survey of Probability Concepts Dr. Stallter has been teaching basic statistics for many years. She knows that 80 percent of the students will complete the assigned problems. She has also determined that among those who do their assignments, 90 percent will pass the course. Among those students who do not do their

### How do you calculate the probability using Bayes Theorem?

One-fourth of the residents of the burning Ridge Estates leave their garage doors open when they are away from home. The local chief of police estimates that 5 percent of the garages with open doors will have something stolen, but only 1 percent of those closed will have something stolen. if a garage is robbed, what is the proba

### Calculating probabilities using Bayes Theorem. Mike Fishbaugh took statistics last semester from Dr.Stallter and received a passing grade. What is the probability that he completed the assignments?

Dr. Stallter has been teaching basic statistics for many years.She knows that 80 percent of the students will complete the assigned problems.She also determined that among those who do their assigments,90 percent will pass the course. Among those students who do not do their homework, 60 percent will pass. Mike Fishbaugh took st

### Discrete Distributions: Chips Being Red

Let X equal the number of "ones" if a fair dice is tossed two independent times by someone out in the hall. If X=x that person gives the player a bowl consisting of 10-3x red chips and 3x white chips. The player then selects one chip at random from the bowl. What is the conditional probability that X=0 given that the chip is red

### Probability and Bayes' Theorem

33. In a survey of MBA students, the following data were obtained on students' first reason for application to the school in which they matriculated. Reason for Application School Quality School cost or convenience Other Total Full Time Part Time 421 400 393 593 76 46 890 1039 Total 821 986 122 1929 a. Develop a joi

### Bayes Rule / Bayesian

Bayesian Analysis of a Law case (taken from Rossman and Short, 1995, Journal of Statistics Education). Joseph Jameison was charged with multiple rapes in 1987 in Allegheny County in the US. DNA evidence from the scenes of the crimes revealed that the attacker displayed the genetic marker PGM2+1- in his DNA. This marker has only

### Probability

Problem #1 A prisoner escapes from jail. There are three roads leading away from the jail. If the prisoner selects road A to make her escape, the probability that she succeeds is 1/4. If she selects road B, the probability that she succeeds is 1/5. If she selects road C, the probability she succeeds is 1/6. Furthermore,

### Probability: Bayesian Statistics

Bayesian Statistics The daily profit from a juice vending machine placed in an office building is a value of a normal random variable with unknown mean μ and variance σ2. Of course, the mean will vary somewhat from building to building, and the distributor feels that these average daily profits can best be described by a n

### Baye's Theorem - Solved Problem

A mail order company employs three stock clerks, A,B and C, who pull items from shelves and assemble them for subsequent verification and packaging. A makes a mistake in an order (gets an item wrong or the wrong quantity) one time in a hundred, B makes a mistake in an order five times in a hundred and C makes a mistake in an or

### Statistics Use of Baye's Theorem, Example solution, Model Solution

Suppose that, in a particular city, airport A handles 50% of all airline traffic, and airports B and C handle 30% and 20%, respectively. The detection rates for weapons at the three airports are .9, .5, and .4, respectively. If a passenger at one of the airports is found to be carrying a weapon through the boarding gate, what is