The most common HIV test is the Enzyme Immunoassay (EIA) Test. When a subject’s HIV condition is known, the following probabilities about the EIA test are known.
Pr(Test(+) I HIV(+)) = 0.9985
Pr(Test(-) I HIV(-)) = 0.994
(a) Find Pr(Test(-) I HIV(+)), the probability of a false negative.
(b) Find Pr(Test(+) I HIV(-)), the probability of a false positive.
(c) Suppose that in the U.S. population, Pr(HIV(1)) = 0.01.
Testing the general population would be effective (both in terms of diagnosis and costs) if a positive test could pinpoint the disease. If a randomly selected person is given the EIA Test, calculate Pr(HIV(+) I Test(+)) and Pr(HIV(-) I Test(+)).
(d) Do the answers from part (c) suggest that testing the general population would be effective? Explain.
A publishing company has determined that a new edition of an existing mathematics textbook will be readopted by 80% of its current users and will be adopted by 7% of the users of other texts if the text is not changed radically. To determine whether it should change the book radically to attract more sales, the company uses Markov chains. Assume that the text in question currently has 25% of its possible market.
(a) Create the transition matrix for this chain.
(b) Find the probability vector for the text three editions later and, from that, determine the percent of the market for that future edition.
(c) Find the steady-state vector for this text to determine what percent of its market this text will have if this policy is continued.
Garage door openers have 10 on-off switches on the opener in the garage and on the remote control used to open the door. The door opens when the remote control is pressed and all the switches on both the opener and the remote agree.
(a) How many different sequences of the on-off switches are possible on the door opener?
(b) If the switches on the remote control are set at random, what is the probability that it will open a given garage door?
(c) Remote controls and the openers in the garage for a given brand of door opener have the controls and openers matched in one of three sequences. If an owner does not change the sequence of switches on her or his remote control and opener after purchase, what is the probability that a different new remote control purchased from this company will open that owner’s door?
(d) What should the owner of a new garage door opener do to protect the contents of his or her garage (and home)?
The new Volvo XC90 Twin-Engine Plug-in Hybrid has a self-programmable mode with four different powertrain settings, three settings for the air suspension, three for the steering, three for the instrument display, two for the brakes, and two for the climate control.
(a) How many differently set cars are possible?
(b) Suppose one owner finds that preference and habit result in narrowing these possibilities to only two of the settings for powertrain, suspension, and steering, just one instrument display, and both settings for brakes and climate control. How many fewer settings does this owner use than are available?
Ideally, auto insurance rates are lower for good drivers than for bad drivers, but an insurer needs to be able to tell which type of driver a client is. Assume that all drivers are considered as either “good drivers” or “bad drivers” and that the probability of a random driver being a “good driver” is 0.80. In addition, suppose that the probability of 2 accidents in a year is 0.02 for good drivers and 0.10 for bad drivers.
(a) If an insurer has no additional information about an applicant’s driving history, what is the probability that the individual is a bad driver?
(b) If the insurer knows that the applicant had 2 accidents in the past year, what is the probability that he or she is a bad driver?
In an actual case,* probability was used to convict a couple of mugging an elderly woman. Shortly after the mugging, a young White woman with blonde hair worn in a ponytail was seen running from the scene of the crime and entering a yellow car that was driven by a Black man with a beard. A couple matching this description was arrested for the crime. A prosecuting attorney argued that the couple arrested had to be the couple at the scene of the crime because the probability of a second couple matching the description was very small. He estimated the probabilities of six events as follows:
Probability of Black-White couple: 1/1000
Probability of Black man: 1/3
Probability of bearded man: 1/10
Probability of blonde woman: 1/4
Probability of hair in ponytail: 1/10
Probability of yellow car: 1/10
He multiplied these probabilities and concluded that the probability that another couple would have these characteristics is 1/12,000,000. On the basis of this circumstantial evidence, the couple was convicted
and sent to prison. The conviction was overturned by the state supreme court because the prosecutor made an incorrect assumption. What error do you think he made?
Would the probabilities for which car he drives 100 days from now depend on whether he drove a Ford or an Audi today?
Use the following information for Problems. A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two days in a row. These are the probabilities that he drives each of the other cars the next day:
Pr(Ford after Audi) = 0.7 Pr(VW after Audi) = 0.3
Pr(Audi after Ford) = 0.6 Pr(VW after Ford) = 0.4
Pr(Audi after VW) = 0.8 Pr(Ford after VW) = 0.2
A certain make of car comes with either an automatic or manual transmission and in one of three styles: convertible, sedan, or SUV. The following table shows the probability of each possible style and transmission.
(a) Find the probability that a car is an automatic.
(b) Find the probability that a car is an automatic given that it is an SUV.
(c) Are “automatic” and “SUV” independent? Explain.
(d) Find the probability that a car is a convertible given that it is a manual.
(e) Are “convertible” and “manual” independent? Explain.
The NAACP claimed that in Volusia County, Florida, police stopped dark-skinned drivers on I-95 who were not violating the law much more frequently than they stopped White drivers and then conducted searches of the cars for drugs or large sums of money that could be related to drug sales. If police made routine stops of randomly selected motorists who were not violating a law and if 22% of the drivers on I-95 passing through this county were dark-skinned, what is the probability that the police stopped a driver who was not violating law but who was dark-skinned? If 39% of the motorists stopped for routine checks were dark-skinned people, did the NAACP claim seem reasonable?
The Pennsylvania Daily Number is a three-digit number game that costs $1 per “number” and has a winning ticket payoff of $500. On April 24, 1980, the winning number was 666. Several weeks later it was discovered that the April 24 drawing had been fixed. Exposure of the scandal revealed that all balls except those with 4 or 6 had been altered so that the winning number would contain only those digits. If you purchased a $1 ticket for each possible 3-digit number that consisted of only 4s or 6s, find the expected payoff on:
(a) April 24, 1980. (First list all possible 3-digit numbers that consist of only 4s or 6s.)
(b) A day when the lottery was not fixed.