12. For a-j below, use the data in the table below which details the value of pleading guilty versus not guilty in criminal courts and the judicial sentencing thereafter. Express all probabilities as decimals rounded to four places (when possible). Guilty Plea Plea of Not Guilty Sentenced to Prison 392, 58. Not Sentenced to Prison 564, 14.

a. If one (1) of the 1028 subjects is randomly selected, find the probability of selecting someone sentenced to prison.

b. Find the probability of being sentenced to prison, given that the subject entered a guilty plea.

c. Find the probability of being sentenced to prison, given that the subject entered a not guilty plea.

d. Based on the results from b & c, what conclusions would you make about the wisdom of entering a guilty plea?

e. If one (1) of the 1028 subjects is randomly selected, find the probability of selecting someone who was sentenced to prison or entered a plea of guilty.

f. If two (2) different study subjects are randomly selected, find the probability that they were both sentenced to prison.

g. Find the probability of randomly selecting 1 of the 1028 subjects and getting a subject who was sentenced to prison and entered a guilty plea.

1. In the United States, the estimated annual probability that a woman over the age of 35 dies of lung cancer is 0.001304 for current smokers and 0.000121 for non-smokers.   

(a) Calculate and interpret the difference in the proportions and the relative risk. Which is more informative for these data? Why? (b) Calculate and interpret the odds ratio. Explain why the relative risk and the odds ratio take on similar values.

2. For people with a particular type of cancer, the odds ratio for recovery (cancer in remission) between age groups (young versus old) was 4.5.   
(a) Choose one correct interpretation:

i. The probability of recovery for young subjects is 4.5 times the probability of recovery for old subjects. ii. The probability of recovery for old subjects is 4.5 times the probability of recovery for young subjects. iii. The odds of recovery for old subjects is 1/4.5 = 0.22 times the odds of recovery for young subjects. iv. The odds of recovery for old subjects is 4.5 times the odds of recovery for young subjects.


(b) Suppose that the odds of remission for young subjects is 3.1. For each age group, find the proportion of subjects who went into remission.

(c) Find the value of R in the interpretation: “The probability of remission for young subjects is R times that of older subjects.”

Probability trees 1. It is estimated that 100 out of 9,900 women who participated in a breast cancer screening had breast cancer. Of the women who participated in the screening and have breast cancer 80 out of 100 will have a positive mammogram. Of the women who participated in the screening and do not have breast cancer 950 out of 9800 will have a positive mammogram.

a) Create a probability tree

b) Calculate the probability that a woman has a positive mammogram given that she does not have breast cancer.

c) Calculate the probability that a woman has a negative mammogram.

d) Given that a woman has a negative mammogram calculate the probability that she has breast cancer.

e) Construct a contingency table comparing cancer diagnosis vs mammogram result.

f) Based on the contingency table what proportion of women have a negative mammogram result given that they have breast cancer.

g) Based on the contingency table what proportion of woman have breast cancer and a positive mammogram test.

h) Calculate the probability that a woman has breast cancer or a positive mammogram result.

i) Can the event “ Cancer Diagnosis or Positive test ” be considered independent events?

Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all four questions, find the indicated probabilities. (Round your answers to three decimal places.) (a) What is the probability that he will answer all questions correctly? (b) What is the probability that he will answer all questions incorrectly? (c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table. Then use the fact that P(r ≥ 1) = 1 − P(r = 0). Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference? They should not be equal, but are equal. They should be equal, but may differ slightly due to rounding error. They should be equal, but differ substantially. They should be equal, but may not be due to table error. (d) What is the probability that Richard will answer at least half the questions correctly?

Richard has just been given a 8-question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all eight questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly?


(b) What is the probability that he will answer all questions incorrectly?


(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.


Then use the fact that P(r ≥ 1) = 1 − P(r = 0).


Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

A: They should not be equal, but are equal.

B: They should be equal, but may not be due to table error.    

C: They should be equal, but differ substantially.

D: They should be equal, but may differ slightly due to rounding error.

(d) What is the probability that Richard will answer at least half the questions correctly?

A tire manufacturer warranties its tires to last at least 20 comma 000 miles or​ "you get a new set of​ tires." In its​ experience, a set of these tires last on average 28 comma 000 miles with SD 5 comma 000 miles. Assume that the wear is normally distributed. The manufacturer profits ​$200 on each set​ sold, and replacing a set costs the manufacturer ​$400. Complete parts a through c.

​(a) What is the probability that a set of tires wears out before 20 comma 000 ​miles? The probability is nothing that a set of tires wears out before 20 comma 000 miles. ​(Round to four decimal places as​ needed.)

​(b) What is the probability that the manufacturer turns a profit on selling a set to one​ customer? The probability is nothing that the manufacturer turns a profit on selling a set to one customer. ​(Round to four decimal places as​ needed.)

​(c) If the manufacturer sells 500 sets of​ tires, what is the probability that it earns a profit after paying for any​ replacements? Assume that the purchases are made around the country and that the drivers experience independent amounts of wear. The probability is nothing that the manufacturer earns a profit after paying for any replacements on 500 sets of tires. ​(Round to four decimal places as​ needed.)

People with albinism have little pigment in their skin, hair, and eyes. The gene that governs albinism has two forms (alleles), which are denoted by a and A. Each person has a pair of these genes, one inherited by from each parent. A child inherits one of each parent’s two alleles, independently with probability 0.5. Albinism is a recessive trait, so a person is albino only of the inherited pair is aa.

Hint: Make a probability tree.

1. Beth’s parents are not albino but she has an albino brother. What types must Beth’s parent’s be AA? Aa? aa?
2. Which of the types AA, Aa, aa, could a child of Beth’s parents have? What is the probability for each type?
3. Beth is not albino. What are the conditional probabilities for Beth’s possible genetic types?

Assume that the probabilities for Beth’s genetic types are given by part (c) above. Beth marries Bob who is albino.

4. What is the conditional probability that a child of Beth and Bob is non-albino if Beth is of type Aa?
5. What is the conditional probability that a child of Beth and Bob is non-albino if Beth is of type AA?
6. Beth and Bob’s first child is non-albino. What is the conditional probability that Beth is of type AA?

Bayes' Rule Problem

Consider the following simplified view of a manager and worker. The worker selects either high (H) or low (L) effort. Given effort, the firm's profit is either x1 or
x2, with x1 < x2.

Assume that the probability of x1, given that the worker selected H, is f(x1 | H) = 1/10 and the probability of x1, given that the worker selected L, is f(x1 | L) = 4/5 . Note
that the f functions are just a formal way of writing conditional probability functions.

As the only possible realizations of x are x1 and x2, we have the probability of x2, given that the worker selected H, is f(x2 | H) = 9/10 and the probability of x2 , given that the worker selected L, is f(x2 | L) = 1/5 .

Assume that the manager's prior probability that the worker selected H is 1/2 .

Assume that, given the profit realization, the manager forms an updated belief about the worker's effort using Bayes' rule.

(a) Suppose the manager observes that x1 is realized. What is the manager's updated belief that the worker selected H?
(b) Suppose the manager observes that x2 is realized. What is the manager's updated belief that the worker selected H?

One state lottery game has contestants select 5 different numbers from 1 to 45. The prize if all numbers are matched is 2 million dollars.   The tickets are $2 each.

1)    How many different ticket possibilities are there?

2)

One state lottery game has contestants select 5 different numbers from 1 to 45. The prize if all numbers are matched is 2 million dollars.   The tickets are $2 each.

1)    How many different ticket possibilities are there?

2)    If a person purchases one ticket, what is the probability of winning? What is the probability of losing?

3)    Occasionally, you will hear of a group of people going in together to purchase a large amount of tickets. Suppose a group of 30 purchases 6,000 tickets.

a)    How much would each person have to contribute?

b)    What is the probability of the group winning? Losing?

   If a person purchases one ticket, what is the probability of winning? What is the probability of losing?

3)    Occasionally, you will hear of a group of people going in together to purchase a large amount of tickets. Suppose a group of 30 purchases 6,000 tickets.

a)    How much would each person have to contribute?

b)    What is the probability of the group winning? Losing?

Agnes Hammer is a senior majoring in management science. She has interviewed with several companies for a job when she graduates, and she is curious about what starting salary offers she might receive. She asked 12 of her classmates at random what their annual staring salary offers were, and she received the following responses

1. Compute the sample mean and sample variance for these data. (Please write down the calculation process)
   

2. Suppose the starting salaries are normally distributed, the mean is the same as the sample mean and variance is the same as sample variance that you calculate from the previous question. What is the probability that Agnes will receive a salary offer of less than $27000? (For probability of normal distribution, please use the probability table posted on Blackboard)
   

3. Suppose the starting salaries are normally distributed, the mean is the same as the sample mean and variance is the same as sample variance that you calculate from the previous question. What is the probability that Agnes will receive a salary offer of between $27000 and $40000 (meaning 27000<= salary <= 40000)? (For probability of normal distribution, please use the probability table posted on Blackboard)