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Chapter 3: 127SE (page 208)

Detecting traces of TNT.University of Florida researchersin the Department of Materials Science and Engineering have invented a technique to rapidly detect traces of TNT (Today, Spring 2005). The method, which involves shining a laser light on a potentially contaminated object, provides instantaneous results and gives no false positives. In this application, a false positive would occur if the laser light detects traces of TNT when, in fact, no TNT is actually present on the object. Let A be the event that the laser light detects traces of TNT. Let B be the event that the object contains no traces of TNT. The probability of a false positive is 0. Write this probability in terms of A and B using symbols such as鈭�,鈭�and |.

Short Answer

Expert verified

The probability through events of A and B is conditional probability.

P(A|B)=0

Step by step solution

01

Important formula

The formula for probability isP=favourableoutcomestotaloutcomes

02

The probability in terms of A and B using symbols such as ∪,∩ and |.

Use characters like union, intersection, and conditional probability to express these probability events A and B.

Now, let鈥檚 recognize the probability, if a laser light identifies traces of TNT, a false positive will arise in this scenario, if there is no TNT existing on the subject. Let A represents the event in which the laser light identifies traces of TNT and B represent the event in which the items have no traces of TNT.

According to the data, a probability of such a false detection is equivalent to that seen in occurrence A in view of occurrence B.

Thus,our event values are the same, this probability through events of A and B is conditional probability.

Therefore, the probability through events of A and B is conditional probability.

P(A|B)=0

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Most popular questions from this chapter

Testing a watch manufacturer鈥檚 claim. A manufacturer of a new Smart Watch claims that the probability of its watch running more than 1 minute slow or 1 minute fast after 1 year of use is .05. A consumer protection agency has purchased four of the manufacturer鈥檚 watches with the intention of testing the claim.

  1. Assuming that the manufacturer鈥檚 claim is correct, what is the probability that none of the watches are as accurate as claimed?
  2. Assuming that the manufacturer鈥檚 claim is correct, what is the probability that exactly two of the four watches are as accurate as claimed?
  3. Suppose that only one of the four tested watches is as accurate as claimed. What inference can be made about the manufacturer鈥檚 claim? Explain.
  4. Suppose that none of the watches tested are as accurate as claimed. Is it necessarily true that the manufacturer鈥檚 claim is false? Explain.

Working mothers with children. The U.S. Census Bureaureports a growth in the percentage of mothers in the workforce who have infant children. The following table gives a breakdown of the marital status and working status of mothers with infant children in the year 2014. (The numbers in the table are reported in thousands.) Consider the following events: A = {Mom with infant works}, B = {Mom with infant is married}. Are A and B independent events?

working

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4064

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2147

1313

(Data from U.S. Census Bureau, Bureau of LabourStatistics, 2014 (Table 4).

Are you really being served red snapper? Red snapper is arare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers

at the University of North Carolina used DNA analysis to examine fish specimens labeled 鈥渞ed snapper鈥� that were purchased from vendors across the country (Nature, July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper but the cheaper, look-alike variety of fish.

a. Assuming the results of the DNA analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant?

b. If there are five customers at a chain restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper?

Advertising proposals. The manager of an advertising department has asked her creative team to propose six new ideas for an advertising campaign for a major client. She will choose three of the six proposals to present to the client. The proposals were named A, B, C, D, E, and F, respectively.

a. In how many ways can the manager select the three proposals? List the possibilities.

b. It is unlikely that the manager will randomly select three of the six proposals, but if she does, what is the probability that she will select proposals A, D, and E?

Two fair coins are tossed, and the following events are defined:

A: [Observe one head and one tail.]

B: [Observe at least one head.]

a. Define the possible sample points and assign probabilities to each.

b. Draw a Venn diagram for the experiment, showing the sample points and events A and B.

c. Find P(A), P(B) andP(A鈭�B).

d. Use the formula for conditional probability to find P (A/B)and P (B/A). Verify your answer by inspecting the Venn diagram and using the concept of reduced sample spaces.
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