### MAT 119 7.6 Bayes Theorem

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TUTORIAL WITH WORK SHOWN FOR THE FOLLOWING QUESTIONS

1. The accompanying tree diagram represents a two-stage experiment. (Let x = 0.5, y = 0.5, r = 0.65, s = 0.35, t = 0.65, and w = 0.35.)

Use the diagram to find the following probabilities. (Round your answers to three decimal places.)

(a) Find P(A) P(D | A).

(b) Find P(B) P(D | B).

(c) Find P(A | D).

1. Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a face card (jack, queen, or king) given that the second card is an ace? (Round your answer to three decimal places.)

3. A halogen desk lamp produced by Luminar was found to be defective. The company has three factories where the lamps are manufactured. The percentage of the total number of halogen desk lamps produced by each factory and the probability that a lamp manufactured by that factory is defective are shown in the accompanying table. What is the probability that the defective lamp was manufactured in Factory III? (Round your answer to three decimal places.)

FactoryPercentage of

Total ProductionProbability of

Defective Component

I330.015

II400.01

III270.02

1. Jansen Electronics has four machines that produce identical components for use in its DVD players. The proportion of the components produced by each machine and the probability of a component produced by that machine being defective are shown in the accompanying table. (Round your answers to three decimal places.)

MachineProportion of

Components ProducedProbability of

Defective Component

I0.150.02

II0.300.02

III0.300.02

IV0.250.04

(a) What is the probability that a component selected at random is defective?

(b) What is the probability that a component selected at random was produced by Machine I, given that it is defective?

(c) What is the probability that a component selected at random was produced by Machine II, given that it is defective?

1. A medical test has been designed to detect the presence of a certain disease. Among people who have the disease, the probability that the disease will be detected by the test is 0.94. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.04. It is estimated that 3% of the population who take this test have the disease. (Round your answers to three decimal places.)

(a) If the test administered to an individual is positive, what is the probability that the person actually has the disease?

(b) If an individual takes the test twice and the test is positive both times, what is the probability that the person actually has the disease? (Assume that the tests are independent.)

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