1. Listed below are the numbers of people implemented in the United States from 1976 to 1994. (In 1976 the U.S. Supreme Court allowed the death penalty to be carried out.)
(See Pagano and Gauvreau, p 32.)
Executions Number of Years
0 – 9 8
10 – 19 5
20 – 29 4
30 – 39 2
i) In how many years were 9 or fewer people implemented?
ii) If you select a year at random, what is the probability that between 10 and 19 people were executed in that year?
iii) If you choose three years at random, what is the probability that in all three of those years between 10 and 19 people were executed? (Be careful; you are sampling without replacement.) (Hint: do the probabilities for the second choice depend on the result of the first choice?)
iv) If you pick three years at random what is the probability that in two of those years between 10 and 19 were executed, and in one of those years at least 30 were executed? (Be careful; you are sampling without replacement.)
v) If you pick three years at random what is the probability that in at least two of those years between 10 and 19 were executed?
2. (Moore and McCabe, third ed.) A life-insurance company sells a term insurance policy to a 21-year-old male. The policy pays $100,000 if the insured male dies within the next five years. The probability that a randomly select male will die every year can be found in mortality tables. The company collects a premium of $250 each year as payment for the insurance. The amount X which the company earns on this policy is the product of $250 and the number of annual payments, less the $100,000 that it should pay if the insured male dies. The table below gives the relevant probabilities. (For instance, if the insured person dies while still 21, the insurance company earns [$250 (from the only premium) $100,000], or – $99,750; the probability of this outcome is 0.00183.)
i) How could we know the relevant probabilities (at least approximately)?
ii) Fill in the missing column.
iii) Compute the expected amount earned through the insurance company from this single policy.
iv) Compute the variance and standard deviation.
v) Compute the relative risk for a single policy.
Age at death 21 22 23 24 25
payout X (in $) -99,750 -99,500 -99250 -99,000 -98,750
probability 0.00183 0.00186 0.00189 0.00191 0.00193
3. (This is a continuation of the previous problem, #2.) assume which you insure two 21-year-old males, and that their ages at death is independent. If we use X and Y for the insurer’s incomes from the two insurance policies, then the total income is T = X + Y. Use the expected value and variance from the previous problem.
i) Compute the expected value, variance, and standard deviation of the total income T.
ii) Determine the relative risk for T.
iii) How many policies should you sell so that the relative risk is 10% of the expected value?
4. At a grocery store, eggs are sold in cartons which contain 12 eggs. Experience suggests that 78.5% of all cartons hold no broken eggs, 19.2% have exactly one broken egg, 2.2% have exactly two broken eggs, and 0.1% have exactly three broken eggs. The fraction of cartons with four or more broken eggs is tiny.
A carton is selected at random, and an egg is selected at random from this carton. This egg is broken.
i) Make a tree diagram illustrating two events: 1) select a carton, and 2) select an egg. Show all the joint outcomes.
ii) Compute the joint probability of selecting a carton with 1 broken egg, and choosing the broken egg from that carton. Express your computation by using the general rule of multiplication.
iii) What are the four joint outcomes in which you choose a broken egg?
iv) Compute the probability of finding at least one broken egg in your carton. Express your computation by using the notation of conditional probability, and the general rule of multiplication.
v) Imagine selecting a carton. Someone looks inside and sees a broken egg. Compute the (conditional) probability that the broken egg is the only broken egg in this carton. (In other words, compute the probability which you have selected a carton with only one broken egg, given that you know there is at least one broken egg in the carton.) Express the computation in notation of conditional probability.
5. The industry journal Oil and Gas discussed following case of prospect fields producing oil (Oil &Gas, January 11, 1988; cited in Anderson, Sweeny, and Williams, Statistics for Business and Economics, 7th edition, p 170). A geological assessment indicated a 25% probability that a particular field would produce oil. There was an 80% probability that a particular well will strike oil, given that oil is present in the prospect field.
i) Make a tree diagram. Demonstrate the selection of a well in the first column, and the result of drilling in subsequent columns.
ii) Assume a well is drilled and it does not strike oil. Compute the probability that the field really does have oil.
iii) If two wells come up dry, what is the probability which the field has oil?
iv) The oil company would like to keep looking as long as the probability which the field has oil is greater than 1%. How many dry wells should be drilled before the field is abandoned?
6. Twenty-nine percent of American judges and lawyers are women (Statistical Abstract of the United States, 1997; cited in Anderson, Sweeney, and Williams, Statistics for Business and Economics, 7th edition, p 197). Imagine a sample of 20 judges and lawyers.
i) Briefly elucidate why you would anticipate the number of number of women in a group of 20 lawyers to be described by a binomial distribution. Refer to the four characteristics of a binomial distribution.
ii) In this illustration, would you anticipate the Poisson distribution to approximately describe the number of women? Why or why not?
iii) Draw a probability histogram, using the binomial tables and a 30% probability-of-success-on-a-single-trial.
iv) Using the binomial-probability formula, find out the probability that 9 of these lawyers and judges are women. Use30% probability-of-success-on-a-single-trial. Contrast your answer to the value in (c) above.
v) Use the binomial tables to find out the probability that at least 9 of the lawyers and judges in the sample of twenty are women.
vi) Compute the expected number of lawyers and judges who are women (in =a sample of 20). Use a 30% probability-of-success-on-a-single-trial. Show your result on the histogram in (c).
vii) Compute the standard deviation of this distribution. Illustrate your result on the histogram in (b). The standard deviation of a binomial distribution is Use a 30% probability-of-success-on-a-single-trial. Show on the histogram in (c).
7. After the Space Shuttle Challenger exploded in 1986, the U.S. Air Force estimated the probability of a “disaster” on a single flight to be 1/34. The Challenger was the 25th flight.
i) Is the number of disasters that happen in a sample of 25 flights describeed by a binomial distribution? Discuss the four characteristics of a binomial distribution from class.
ii) NASA kept flying shuttle missions until the first disaster occurred. Is the number of safe missions in THIS case described by a binomial distribution? describe.
iii) Compute the probability that one flight in a group of 25 will end in disaster.
iv) Compute the probability that exactly two flights (in a group of 25 flights) end in disaster.
v) Compute the probability of AT LEAST one disaster in 25 flights.
vi) Create a binomial probability histogram for the number of disastrous flights in a sample of 25. (Use a table of binomial probabilities.)
8. Refer to problem #7.
i) Compute the expected number of disasters in 25 flights.
ii) Imagine that the probability of a specific number of disasters is described by a Poisson distribution. Use your result in (c) to find out the probability of at least one disaster in 25 flights. Contrast your answer to #7e above.
iii) describe why the number of disasters that occur in a sample of 25 flights should be described approximately by a Poisson distribution.
iv) Create a Poisson probability histogram for the number of disastrous flights in a sample whose expected value is given in (a) above.
v) Is the histogram in (d) approximately the same as the histogram in problem #7 above?
9. The number of cases of diptheria reported in United States in any particular year between 1980 and 1987 is describeed by a Poisson distribution whose mean is 2.5 (Centers for Disease Control, “Summary of Notifiable Diseases, United States, 1989”, Morbidity and Mortality Weekly Report, Volume 39, October 5, 1990; cited in Pagano and Gauvreau, Principles of Biostatistics, 1st edition, p 170).
i) Compute the probability that exactly four cases are reported in a particular year. Employ both the Poisson tables and Poisson formula.
ii) Compute the probability that no more than four cases are reported in a particular year.
iii) Draw TWO probability histograms for the number of diptheria cases in any particular year.
(a) Employ one of the histograms to illustrate the probability in (a) (i.e. the probability that exactly four cases are reported in a particular year).
(b) Employ the second histogram to describe the probability in (b) (i.e. the probability that no more than four cases are reported in a particular year).
iv) Compute the distribution’s standard deviation. Show the mean and standard deviation on one of the histograms in (c).
10. In a particular class, 65% of the students are men. The students have just written a test, and 55% of the students passed. Imagine randomly choosing one of the students: the probability that this student both passed and is male is 22%.
i) Make a contingency table to display this information.
ii) Randomly select a student: the student turns out to be male. Compute the probability that a student passed the test, given that the student is male.
iii) Randomly select a student; this student passed this test. Compute the probability that a student is male, given that the student passed the test.
iv) Compute the probability that a randomly chosen student is a woman, given that the student passed.
v) Compute the probability that a randomly chosen student passed the test, given that the student is a woman.
vi) Are the test results and the gender of the students independent? Clearly describe your reasoning.
11. I have oversimplified the following situation, in order (I hope) to make a point.
Assume the following:
• A crime is committed in a city of one million adults;
• The guilty party’s DNA was found at the crime scene;
• The probability that DNA taken from the actual criminal will be misread is zero (or the test’s SENSITIVITY = 1);
• The probability that a randomly chosen person’s DNA will apparently match the criminal’s DNA is 1 in a million (or the test’s SPECIFICITY = 999,999/1,000,000);
• The DNA of a suspect actually in custody matches the criminal’s DNA;
• The crown prosecutor alleges that, since the probability that the suspect’s DNA could, by chance, match the criminal’s DNA is 1 in a million, the probability that the suspect is innocent is 1 in a million.
i) Imagine testing ALL 1 million adults in this city (somewhat extreme and unrealistic). How many false positives would you expect?
ii) How many right positives would you anticipate? (Hint: there is only one criminal.)
iii) What fraction of the people who test positive (in (b) above) are guilty?
iv) find out the probability that a suspect is guilty, given a positive test result.
v) Express what the prosecutor has claimed in the language of conditional probability.