Go to the web and find more about the history of 2 probability theorists

mentioned in this chapter.

3. Using diagrams, convince yourself of the rationality of De Morgan's law:

(A ? B)

c = Ac ? Bc (1.66)

4. Try to prove analytically De Morgan's law.

5. Urn A has 3 black balls and 6 white balls. Urn B has 400 black balls and

400 white balls. Urn C has 6 black balls and 3 white balls. A person first

randomly chooses one of the urns and then grabs a ball randomly from the

chosen urn. What is the probability that the ball be black? If a person

grabbed a black ball. What is the probability that the ball came from urn B?

6. The probability of catching Lyme disease after on day of hiking in the Cuyamaca mountains are estimated at less than 1 in 10000. You feel bad after a

day of hike in the Cuyamacas and decide to take a Lyme disease test. The

test is positive. The test specifications say that in an experiment with 1000

patients with Lyme disease, 990 tested positive. Moreover. When the same

test was performed with 1000 patients without Lyme disease, 200 tested

positive. What are the chances that you got Lyme disease.

7. This problem uses Bayes' theorem to combine probabilities as subjective

beliefs with probabilities as relative frequencies. A friend of yours believes

she has a 50% chance of being pregnant. She decides to take a pregnancy

test and the test is positive. You read in the test instructions that out of

100 non-pregnant women, 20% give false positives. Moreover, out of 100

pregnant women 10% give false negatives. Help your friend upgrade her

beliefs.

8. In a communication channel a zero or a one is transmitted. The probability

that a zero is transmitted is 0.1. Due to noise in the channel, a zero can be

received as one with probability 0.01, and a one can be received as a zero

with probability 0.05. If you receive a zero, what is the probability that a

zero was transmitted? If you receive a one what is the probability that a one

was transmitted?

9. Consider a probability space (?, F, P). Let A and B be sets of F, i.e., both

A and B are sets whose elements belong to ?. Define the set operator $$

20) Playbill is a magazine distributed around the country to people attending musicals and other theatrical productions. The mean annual household income for the population of Playbill reader is $119,155. Assume the standard deviation is a = $20,700, A San Francisco civic group has asserted that the mean for theatergoers in the Bay Area is higher. A sample of 60 theater attendees in the Bay Area showed a sample mean household income of $126,100. a) Develop hypotheses that can be used to determine whether the sample data support the conclusion that theater attendees in the Bay Area have a higher mean household income than that for all Playbill readers. b) What is the p-value based on the sample of 60 theater attendees in the Bay Area? art c) Use a = 0.01 as the level of significance. What is your conclusion?t Direction - gn Text Shape Outline ~ 4ac Replace Arrange Quick nvert to SmartArt - Styles Shape Effects Select Drawing Editing Bayes' Rule ayes' Rule (General Case): Let A1,..., A, be mutually exclusive and exhaustive events, with P(A;) # 0 for each A. Let B be any event with P(B) #0. Then P(A | B) P(B | A ) P(A.) n [P( B | A,) P(A ) i=1 PERCENTEPart B. For some events C and D, suppose that P(C) = 0.2, P(D) = 0.4, and P(C n D) = 0.1. Can C and D be independent events? O Maybe O Yes O No O Cannot be determined based on the information given Can C and D be disjoint events? O Maybe O No O Cannot be determined based on the information given O Yes Using the definition of the conditional probability, find the following probabilities: P(C | D) = P(D | C) = Part C. For some events E and F, suppose that P(F) = 0.3, P(E | F) = 0.4, and P(E | FC) = 0.2. To find P(E / F), what probability rule/law, should you use? O The law of total probability O Axioms of probability O Definition of the conditional probability O The addition rule O Definition of the independent events O The multiplication rule for P(A / B) O Bayes' theorem Find P(E n) F). To find P(E), what probability rule/law, should you use? O Bayes' theorem O Axioms of probability O Definition of the conditional probability O The law of total probability O Definition of the independent events O The multiplication rule for P(A / B) O The addition rule Find P(E)