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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%!

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

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

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

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

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


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,

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

Bayes Right-Winger Committee

A Committee of three people has been formed by random selection from five left-wingers and four right wingers. The Committee members then vote for or against a strike whenever there is a dispute. Each left winger votes for a strike three out of four times in strike votes, whereas each right winger votes for a strike only once ou

Bayes Analysis Problem

Assume that a patient is believed to have one of two diseases, denoted by D1 and D2 with P(D1)=0.40 and P(D2)=0.60 and that medical research has determined the probability associated with each symptom that may accompany the diseases. Suppose that given diseases D1 and D2, the probabilities that the patient will have symptom S1,

Probability and Business Decisions

a. See the attached file for the chart of data pertaining to this question. The probabilities shown above represent the states of nature and the decision maker's (e.g., manager) degree of uncertainties and personal judgment on the occurrence of each state. What is the expected payoff for actions A1 and A2 above? What would b

Bayesian estimate

Suppose theta is a parameter that takes values on the real line. Consider the loss function l(theta, theta_hat) = 0 if |q-theta_hat|<=c, 1 otherwise Show that the Bayes estimate of theta is the midpoint of the interval I of length 2c that maximizes P(theta in I|x).

Probabilities: Answer the following questions and show all your calculations.

Over the 48 years from 1950 through 1997, the stock market has gone up in the month of January for 31 times; it has gone up for the whole year for 36 times, and it has gone up both for the year and in January for 29 times. 1) Based on historical data, what is the probability the stock market will go up in January? 2) B

Probability - Baye's Theorem

Question: Urn I has 2 white and 3 black balls; Urn II, 4 white and 1 black; and Urn III, 3 white and 4 black. An urn is selected at random and a ball drawn at random is found to be white. Find the probability that Urn I was selected. Answer: 14/57


Category: Statistics Subject: probabilities and Bayes theorem Details: 1. (Independence of events). The chancellor of a state university is applying for a new position. At a certain point in his application process, he is being considered by seven universities. At three of the seven he is a finalist, which means that he

Working with Bayes' theorem calculations.

Experience has shown a company that the cost of delivering a small package within 24 hours is $14.80. The company charges $15.50 for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 2% of its packages within the 24-hour period, what is the expected gain