Chapter Five Review—Answers For Questions 1–8, answer true or false.

1. Two events, each with probability greater than 0, are mutually exclusive (disjoint). The probability that both occur on the same opportunity is 0. True

2. You have flipped a fair coin and got five heads in a row. The probability that you will get heads on the next flip is less than 0.5. False

3. When sampling units randomly from a population with replacement, pairs of successive selections are independent. True

4. If events A and B are disjoint, then they are independent. False

5. The sample space for randomly selecting two people to form a team from a group of five people contains ten equally likely outcomes. True

a. From these 5 ppl: ABCDE: teams are AB AC AD AE BC BD BE CD CE DE

6. An event A and its complement A must be mutually exclusive (disjoint). True

7. If you roll two dice, all possible sums are equally likely. False

8. If events A and B are independent, then P(A and B) = P(A) + P(B). False. That’s P(A or B).

9. Armine and her friend Terry both hope to get an A in math this year. Their teacher estimates the probability that Armine gets an A is 0.7 and that Terry gets an A is 0.8. If the probability that one or both gets an A is 0.9, what is the probability that both get an A? A. 0.34 B. 0.56 C. 0.60 D. 0.90 E. We cannot determine the probabilities unless we know that Armine gets an A and Terry gets an A are independent

events.

Questions 10-13 refer to this study: A sociologist was interested in studying the relationship between how long an employee commutes and whether the employee works full- or part-time. This table shows that information for a total of 2000 employees. An employee is selected at random from this group.

Commuting Time

Less Than 30 Minutes

Between 30 and 60 Minutes

Greater Than 60 Minutes Total

Employment Status Part-Time 540 440 240 1220

Full-Time 410 260 110 780

Total 950 700 350 2000

10. What is the probability that an employee chosen at random works full-time and commutes less than 30 minutes? 410/2000 A. 0.205 B. 0.39 C. 0.475 D. 0.66 E. 0.865

11. What is the probability that an employee chosen at random works full-time or commutes less than 30 minutes? (410+260+110+540)/2000 A. 0.205 B. 0.39 C. 0.475 D. 0.66 E. 0.865

12. What is the probability that an employee chosen at random works part-time given that he or she commutes more than 60 minutes? 240/350 A. 0.12 B. 0.20 C. 0.31 D. 0.69 E. 0.785

13. Are the events commutes more than 60 minutes and works full-time independent? A. No, because P(commutes more than 60 minutes | works full-time) ≠ P(commutes more than 60 minutes). B. No, because P(commutes more than 60 minutes | works full-time) ≠ P(works full-time). C. Yes, because P(commutes more than 60 minutes | works full-time) ≠ P(commutes more than 60 minutes). D. Yes, because P(commutes more than 60 minutes | works full-time) ≠ P(works full-time). E. No, because some full-time employees commute more than 60 minutes.

14. Suppose that the probability that a student selected at random takes statistics is 0.35, the probability that a student selected at random takes both statistics and biology is 0.19, and the probability that a student selected at random takes statistics but not biology is 0.17. Which of these is a proper conclusion? A. The probability that the student takes biology is 0.36. B. The probability that the student doesn’t take biology is 0.21. C. The probability that the student takes statistics or biology is 0.71. D. The probability that the student takes biology but not statistics is 0.18. E. None of the above is true because the probabilities given in the question are contradictory.

Stats No Stats

Bio .19

No Bio .17

.35≠.19+.17

15. A recent census found that 40% of students are against the idea of requiring students to carry identification cards (IDs) on campus. Of those against the ID requirement, 60% were seniors, 30% were juniors, and the remaining 10% were underclassmen (freshmen and sophomores). Of those in favor of IDs, 20% were seniors and 50% were underclassmen.

a. Construct a table or draw a tree diagram that summarizes this situation. SRs JRs Underclass Totals Against IDs .24 .12 .04 .4 In Favor of IDs .12 .18 .3 .6 Totals: .36 .3 .34 1

b. What is the probability that a student chosen at random is a senior? .36 c. What is the probability that a senior chosen at random is in favor of the ID requirement? P(in Favor of IDs | Sr) = .12/.36 = 1/3 or .33

16. Consider a screening test for Rocky Mountain Fever that has reasonably good specificity and sensitivity: Of people who have the illness, 95% get a positive result on the test, and of people who don’t have the illness, 92% get a negative result on the test. Now consider Rockytown, which has a population of 10,000 people; in the town, 100 people have Rocky Mountain Fever. a. Fill out a table or draw a tree diagram to show how the test would perform if you used it to screen the residents of

Rockytown. (see below, 2-way table and tree diagram)

RMF No RMF Totals Test Pos 95 792 887 Test Neg 5 9108 9113 Totals 100 9900 10,000

b. Compute the positive predictive value (probability of having the disease given the test is positive).

𝑷𝑷(𝑹𝑹𝑹𝑹𝑹𝑹|𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻 +) =𝑷𝑷(𝑹𝑹𝑹𝑹𝑹𝑹 & 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻+)

𝑷𝑷(𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻+)=

𝟗𝟗𝟗𝟗𝟖𝟖𝟖𝟖𝟖𝟖

𝒐𝒐𝒐𝒐 .𝟎𝟎𝟎𝟎𝟗𝟗𝟗𝟗

.𝟎𝟎𝟎𝟎𝟗𝟗𝟗𝟗+.𝟎𝟎𝟖𝟖𝟗𝟗𝟎𝟎=.𝟏𝟏𝟎𝟎𝟖𝟖

c. Compute the negative predictive value (probability of not having the disease given the test is negative).

𝑷𝑷(𝑵𝑵𝒐𝒐 𝑹𝑹𝑹𝑹𝑹𝑹|𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻 −) =𝑷𝑷(𝑵𝑵𝒐𝒐 𝑹𝑹𝑹𝑹𝑹𝑹 & 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻−)

𝑷𝑷(𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻−)=𝟗𝟗𝟏𝟏𝟎𝟎𝟖𝟖𝟗𝟗𝟏𝟏𝟏𝟏𝟗𝟗

𝒐𝒐𝒐𝒐 .𝟗𝟗𝟏𝟏𝟎𝟎𝟖𝟖

.𝟗𝟗𝟏𝟏𝟎𝟎𝟖𝟖+.𝟎𝟎𝟎𝟎𝟎𝟎𝟗𝟗=.𝟗𝟗𝟗𝟗𝟗𝟗

d. Comment on these rates and what they mean for the usefulness of the test. Due in large part to the scarcity of Rocky Mountain Fever (1/100 of the population), the positive predictive

value (probability of having RMF given that the test was positive) is very small, about 11%. So the test is only useful 11% of the time it indicates a positive result.

By contrast, the test’s negative predictive value is 99.9%. So it is a VERY useful predictor of the absence of Rocky Mountain Fever.

17. The proportion of voters who voted for George W. Bush in the 2000 presidential election was approximately 0.48. a. Describe how to use these lines from a random digit table to simulate taking a sample of 10 people who voted in the

2000 presidential election and recording how many of the 10 voted for Bush.

11805 05431 39808 27732 50725 68248

83452 99634 06288 98083 13746 70078

88685 40200 86507 58401 36766 67951

Use the digits in pairs, each pair will represent one voter. If the digit pair is 00-47, then the person voted for Bush. If the digits are 48-99, then the person did not vote for Bush. One trial of the simulation will be 10 pairs (10 voters). At the end of the trial, record how many of the 10 voted for Bush.

b. Start at the beginning of the first line, take three samples of size 10, and estimate the probability that no more than 3 of the 10 voted for Bush. (Do not start a new line for each sample. Start where the previous sample finished.)

Trial 1: 4/10 voters for Bush

Trial 2: 3/10 voters for Bush

Trial 3: 4/10 voters for Bush

P(no more than 3 of the 10 voted for Bush) = 1/3 or .333 or 33.3%

Rockytown

RMF (.01)Test + (.95) True Pos

(.0095)

Test - (.05) False Neg (.0005)

No RMF (.99)

Test + (.08) False Pos (.0792)

Test - (.92) True Neg (.9108)

According to the simulation of 3 trials, the estimated probability that no more than 3 of 10 voters voted for Bush is approximately 33.3%.

18. A new clothing store advertises that during its Grand Opening every customer that enters the store can throw a bouncy rubber cube onto a table that has squares labeled with discount amounts. The table is divided into ten regions. Five regions award a 10% discount, two regions award a 20% discount, two regions award a 30% discount, and the remaining region awards a 50% discount.

a. What is the probability that a customer gets more than a 20% discount?

b. What is the probability that a customer gets less than a 20% discount?

c. What is the probability that the first two customers both get a 50% discount?

d. What is the probability that none of the first three customers gets more than a 30% discount?

e. What is the probability that the first customer to win a 30% discount is the sixth customer that enters the store?

f. What is the probability that there is at least one customer to win a 50% discount among the first five customers that

enter the store?

g. As you enter the store you watch the four people in front of you all win 50% discounts. The store manager tells you

how lucky you are to be throwing the cube while it is on a hot streak, but the friend with you says you’re unlucky because the streak can’t continue. Comment on their statements.

The tosses are independent, so if the table and cube are fair the four winners in front have no effect on the next person’s chances of tossing a 50% discount.