1. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an A for the semester. Your subjective assessment of your performance is Event Probability A on paper and A on exam .25 A on paper only .10 A on exam only .30 A on neither .35 Let's develop a joint-probability table to analyze this problem (the letters a – h represent probability values. For example, a is the probability of getting an A on both the paper and the exam; b is the probability of getting an A on the exam, but not on the paper; while g is the sum of a and d, and represents the probability of getting an A on the paper). Get A on paper Do not get A on paper row sum Get A on exam a b c Do not get A on exam d e f column sum g h
a. What does c + f equal?
b. What does a + b + d + e equal?
c. What is the probability of getting an A on the exam?
d. What is the probability of getting an A in the course?
e. Are the grades on the assignments independent (explain your answer mathematically)?
2. A medical research project examined the relationship between a subject’s weight and recovery time from a surgical procedure, as shown in the table below. Underweight Normal weight Overweight Less than 3 days 6 15 3 3 to 7 days 30 95 20 Over 7 days 14 40 27
a. Use relative frequency to develop a joint probability table to show the marginal probabilities.
b. What is the probability a patient will recover in fewer than 3 days?
c. Given that recovery takes over 7 days, what is the probability the patient is overweight?
3. The Ambell Company uses batteries from two different manufacturers. Historically, 60% of the batteries are from manufacturer 1, and 90% of these batteries last for over 40 hours. Only 75% of the batteries from manufacturer 2 last for over 40 hours. A battery in a critical tool fails at 32 hours. What is the probability it was from manufacturer 2?
4. The high school GPA of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories. GPA f(x) x < 2.0 .08 2.0 <= x < 2.5 .12 2.5 <= x < 3.0 .35 3.0 <= x < 3.5 .30 3.5 <= x ?
a. Complete the table to make this a valid probability distribution.
b. What is the probability an applicant's GPA will be below 3.0?
c. What is the probability an applicant's GPA will be 2.5 or above?
5. A video rental store has two video cameras available for customers to rent. Historically, demand for cameras has followed this distribution. The revenue per rental is $40. If a customer wants a camera and none is available, the store gives a $15 coupon for tape rental. Demand Relative Frequency Revenue Cost 0 .35 0 0 1 .30 40 0 2 .20 80 0 3 .10 80 15 4 .05 80 30
a. What is the expected demand?
b. What is the expected revenue?
c. What is the expected cost?
d. What is the expected profit?
6. A manufacturer of computer disks has a historical defective rate of .001. What is the probability that in a batch of 1000 disks, 2 would be defective? (note: answer using either the relevant probability table in the back of the book, or use the relevant probability function on your calculator / Excel).
7. After a severe winter, potholes develop in a state highway at the rate of 5.2 per mile. Thirty-five miles of this highway pass through Washington County.
a. How many potholes would you expect to see in the county?
b. What is the probability of finding 8 potholes in 1 mile of highway?
8. The weight of a can of stewed tomatoes is listed as 16 oz., but in actuality, the weight between cans may vary, and in fact is normally distributed with mean = 16.2 ounces and standard deviation = 0.1 oz. What is the probability that a consumer buys a can whose contents are actually less than the stated weight of 16 oz.?