MAT 119 7.5 Conditional Probability and Independent Events

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TUTORIAL WITH WORK SHOWN FOR THE FOLLOWING QUESTIONS

  1. Let A and B be two events in a sample space S such that P(A) = 0.5, P(B) = 0.5, and P(A ? B) = 0.15. Find the probabilities below.

(a) P(A|B)

(b) P(B|A)

  1. If A and B are independent events, P(A) = 0.25, and P(B) = 0.65, find the probabilities below. (Enter your answers to four decimal places.)

(a) P(A ? B) =

(b) P(A B) =

(c) P(A | B) =

(d) P(Ac Bc) =

  1. The accompanying tree diagram represents an experiment consisting of two trials.

Use the diagram to find the probabilities below.

(a) P(A) =

(b) P(E | A) =

(c) P(A E) =

(d) P(E) =

(e) Does P(A E) = P(A) P(E)?

(f) Are A and E independent events?

  1. An experiment consists of two independent trials. The outcomes of the first trial are A, B, and C, with probabilities of occurring equal to 0.2, 0.6, and 0.2, respectively. The outcomes of the second trial are E and F, with probabilities of occurring equal to 0.4 and 0.6. Draw a tree diagram representing this experiment. Use this tree diagram to find the probabilities below.

(a) P(B) =

(b) P(F | B) =

(c) P(B F) =

(d) P(F) =

(e) Does P(B F) = P(B) P(F)?

(f) Are B and F independent events?

  1. A tax specialist has estimated that the probability that a tax return selected at random will be audited is 0.05. Furthermore, he estimates the probability that an audited return will result in additional assessments being levied on the taxpayer is 0.30. What is the probability that a tax return selected at random will result in additional assessments being levied on the taxpayer?

  2. In a survey of 1000 eligible voters selected at random, it was found that 200 had a college degree. Additionally, it was found that 70% of those who had a college degree voted in the last presidential election, whereas 46% of the people who did not have a college degree voted in the last presidential election. Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random will have the following characteristics. (Round your answers to three decimal places.)

(a) The voter had a college degree and voted in the last presidential election.

(b) The voter did not have a college degree and did not vote in the last presidential election.

(c) The voter voted in the last presidential election.

(d) The voter did not vote in the last presidential election.

  1. An automobile manufacturer obtains the microprocessors used to regulate fuel consumption in its automobiles from three microelectronic firms: A, B, and C. The quality-control department of the company has determined that 5% of the microprocessors produced by firm A are defective, 6% of those produced by firm B are defective, and 5.5% of those produced by firm C are defective. Firms A, B, and C supply 35%, 30%, and 35%, respectively, of the microprocessors used by the company. What is the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor?

  2. The chief loan officer of La Crosse Home Mortgage Company summarized the housing loans extended by the company in 2010 according to type and term of the loan. Her list shows that 71% of the loans were fixed-rate mortgages (F), 23% were adjustable-rate mortgages (A), and 6% belong to some other category (O) (mostly second trust-deed loans and loans extended under the graduated payment plan). Of the fixed-rate mortgages, 42% were 30-year loans, and 58% were 15-year loans; of the adjustable-rate mortgages, 30% were 30-year loans and 70% were 15-year loans; finally, of the other loans, extended, 34% were 20-year loans, 60% were 10-year loans, and 6% were for a term of 5 years or less.

(a) Draw a tree diagram representing these data.

(b) What is the probability that a home loan extended by La Crosse has an adjustable rate and is for a term of 15 years? (Round your answer to four decimal places.)

(c) What is the probability that a home loan extended by La Crosse is for a term of 15 years? (Round your answer to four decimal places.)

  1. The personnel department of Franklin National Life Insurance compiled the accompanying data regarding the income and education of its employees.

Income

$60,000 or Below    Income

Above $60,000    

Noncollege Graduate 2060820

College Graduate370750

Let A be the event that a randomly chosen employee has a college degree, and let B be the event that the chosen employee's income is more than $60,000.

(a) Find each of the following probabilities. (Round your answers to four decimal places.)

(b) Are the events A and B independent events?

  1. According to a survey conducted in 2004 of 1000 American adults with Internet access, one in five households planned to switch ISPs in the next 6 months. Of those who planned to switch, 3% of the households were likely to switch to a satellite connection, 30% to a digital subscriber line (DSL), 26% to cable modem, 33% to dial-up modem, and 8% didn't know what kind of service provider they would switch to. (Round your answers to three decimal places.)

(a) What is the probability that a randomly selected survey participant switched to a dial-up modem connection?

(b) What is the probability that a randomly selected survey participant who was planning to switch ISPs upgraded to high-speed service (satellite, DSL, or cable)?

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