Test 1 (ch. 4) Practice Questions

(4.1#21)Sometimes probability statements are expressed in terms of odds. (LOOK AT TEST 1 PICTURE REFERENCE) For instance, if P(A)=0.60, then P(A^c)=0.40 and the odds in favor of A are 3:2 (b)A telemarketing supervisor tells a new worker that the odds of making a sale on a single call are 2 to 15. What is the probability of a successful call? (c)A sports announcer says that the odds a basketball player will make a free throw shot are 3 to 5. What is the probability the player will make the shot?

(b) P(success) 5 2/17 <0.118. (c) P(make shot) 5 3/8 or 0.375

(4.2#35)The state medical school has discovered a new test for tuberculosis. (If the test indicates a person has tuberculosis, the test is positive.) Experimentation has shown that the probability of a positive test is 0.82, given that a person has tuberculosis. The probability is 0.09 that the test registers posi-tive, given that the person does not have tuberculosis. Assume that in the general population, the probability that a person has tuberculosis is 0.04. What is the probability that a person chosen at random will (a)have tuberculosis and have a positive test? (b)not have tuberculosis? (c)not have tuberculosis and have a positive test?

(a) 0.033 (b) 0.96 (c) 0.086

(4.2#5)Given P(A)=0.2 and P(B)=0.4: (a)If A and B are independent events, compute P(A and B). (b)If P(A | B)=0.1, compute P(A and B).

(a) 0.08. (b) 0.04.

(4.2#7)Given P(A)=0.2, P(B)=0.5, P(A | B)=0.3: (a)Compute P(A and B). (b)Compute P(A or B).

(a) 0.15. (b) 0.55.

(4.2#15)M&M plain candies come in various colors. The distribution of colors for plain M&M candies in a custom bag is (LOOK AT TEST 1 PICTURE REFERENCE) Suppose you have a large custom bag of plain M&M candies and you choose one candy at random. Find (a)P(green candy or blue candy). Are these outcomes mutually exclusive? Why? (b)P(yellow candy or red candy). Are these outcomes mutually exclusive? Why? (c)P(not purple candy)

(a) 0.2; yes. (b) 0.4; yes. (c) 1.0 - 0.2 = 0.8.

(4.2#25)USA Today gave the information shown in the table about ages of children receiving toys. The percentages represent all toys sold. (LOOK AT TEST 1 PICTURE REFERENCE) What is the probability that a toy is purchased for someone (a)6 years old or older? (b)12 years old or younger? (c)between 6 and 12 years old? (d)between 3 and 9 years old? A child between 10 and 12 years old looks at this probability distribution and asks, "Why are people more likely to buy toys for kids older than I am [13 and over] than for kids in my age group [10-12]?" How would you respond?

(a) 0.63 (b) 0.78 (c) 0.41 (d) 0.49 The category 13 and over contains far more ages than the group 10-12. It is not surprising that more toys are purchased for this group, since there are more children in this group.

(4.2#33)Wing Foot is a shoe franchise commonly found in shopping centers across the United States. Wing Foot knows that its stores will not show a profit unless they gross over $940,000 per year. Let A be the event that a new Wing Foot store grosses over $940,000 its first year. Let B be the event that a store grosses over $940,000 its second year. Wing Foot has an administrative policy of closing a new store if it does not show a profit in either of the first 2 years. The accounting office at Wing Foot provided the following information: 65% of all Wing Foot stores show a profit the first year; 71% of all Wing Foot stores show a profit the second year (this includes stores that did not show a profit the first year); however, 87% of Wing Foot stores that showed a profit the first year also showed a profit the second year. Compute the following: (a)P(A) (b)P(B) (c)P(B | A) (d)P(A and B) (e)P(A or B) (f)What is the probability that a new Wing Foot store will not be closed after 2 years? What is the probability that a new Wing Foot store will be closed after 2 years?

(a) 0.65 (b) 0.71 (c) 0.87 (d) 0.57 (e) 0.79 (f) P(not close)= 0.79; P(close)= 0.21

(4.2#3)Given P(A)=0.3 and P(B)=0.4: (a)If A and B are mutually exclusive events, compute P(A or B). (b)If P(A and B)=0.1, compute P(A or B).

(a) 0.7. (b) 0.6.

(4.1#3)What is the probability of (a) an event A that is certain to occur? (b) an event B that is impossible?

(a) 1. (b) 0

(4.2#19)You roll two fair dice, a green one and a red one. (a)What is the probability of getting a sum of 6? (b)What is the probability of getting a sum of 4? (c)What is the probability of getting a sum of 6 or 4? Are these outcomes mutually exclusive?

(a) 5/36 (b) 3/36 (c) 8/36

(4.2#29) In a sales effectiveness seminar, a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach. For 1160 customers, the following record was kept: (LOOK AT TEST 1 PICTURE REFERENCE) Suppose a customer is selected at random from the 1160 participating custom-ers. Let us use the following notation for events: A=aggressive approach, Pa=passive approach, S=sale, N=no sale. So, P(A) is the probability that an aggressive approach was used, and so on. (a)Compute P(S), P(S | A), and P(S | Pa). (b)Are the events S=sale and Pa=passive approach independent? Explain. (c)Compute P(A and S) and P(Pa and S). (d)Compute P(N) and P(N | A). (e)Are the events N=no sale and A=aggressive approach independent? Explain. (f)Compute P(A or S).

(a) 686/1160; 270/580; 416/580. (b) No. (c) 270/1160; 416/1160. (d) 474/1160; 310/580. (e) No. (f ) 686/1160 + 580/1160 - 270/1160 = 996/1160.

(4.2#31)In an article titled "Diagnostic accuracy of fever as a measure of postoperative pulmonary complications" (Heart Lung, Vol. 10, No. 1, p.61), J. Roberts and colleagues discuss using a fever of 38*C or higher as a diagnostic indicator of postoperative atelectasis (collapse of the lung) as evidenced by x-ray observation. For fever >or=38*C as the diagnostic test, the results for post-operative patients are (LOOK AT TEST 1 PICTURE REFERENCE) (a)P(+ | condition present); this is known as the sensitivity of a test. (b)P(- | condition present); this is known as the false-negative rate. (c)P(- | condition absent); this is known as the specificity of a test. (d)P(+ | condition absent); this is known as the false-positive rate.(e)P(condition present and +); this is the predictive value of the test.(f)P(condition present and -)

(a) 72/154. (b) 82/154. (c) 79/116. (d) 37/116. (e) 72/270. (f ) 82/270.

(4.3#27)The qualified applicant pool for six management trainee positions consists of seven women and five men. (a)How many different groups of applicants can be selected for the positions? (b)How many different groups of trainees would consist entirely of women? (c)Probability Extension If the applicants are equally qualified and the trainee positions are selected by drawing the names at random so that all groups of six are equally likely, what is the probability that the trainee class will consist entirely of women?

(a) 924 (b) 7 (c) 0.008

(4.1#23)John runs a computer software store. Yesterday he counted 127 people who walked by his store, 58 of whom came into the store. Of the 58, only 25 bought something in the store. (a)Estimate the probability that a person who walks by the store will enter the store. (b)Estimate the probability that a person who walks into the store will buy something. (c)Estimate the probability that a person who walks by the store will come in and buy something. (d)Estimate the probability that a person who comes into the store will buy nothing.

(a) P(enter if walks by) = 58/127 <0.46. (b) P(buy if entered)= 25/58 <0.43. (c) P(walk in and buy) = 25/127 <0.20. (d) P(not buy) = 1 -P(buy) = 1 - 0.43 = 0.57

(4.3#5) (a)Draw a tree diagram to display all the possible head-tail sequences that can occur when you flip a coin three times. (b)How many sequences contain exactly two heads? (c)Probability Extension Assuming the sequences are all equally likely, what is the probability that you will get exactly two heads when you toss a coin three times?

(a) SEE BACK BOOK (b) 3. (c) 3/8.

(4.3#7)There are six balls in an urn. They are identical except for color. Two are red, three are blue, and one is yellow. You are to draw a ball from the urn, note its color, and set it aside. Then you are to draw another ball from the urn and note its color. (a)Make a tree diagram to show all possible outcomes of the experiment. Label the probability associated with each stage of the experiment on the appropriate branch. (b)Probability Extension Compute the probability for each outcome of the experiment.

(a) SEE BACK BOOK (b) P(R and R) = 2/6 * 1/5 = 1/15. P(R 1st and B 2nd) = 2/6 * 3/5 = 1/5. P(R 1st and Y 2nd) = 2/6 * 1/5 = 1/15. P(B 1st and R 2nd) = 3/6 * 2/5 = 1/5. P(B 1st and B 2nd) = 3/6 * 2/5 = 1/5. P(B 1st and Y 2nd) = 3/6 * 1/5 = 1/10. P(Y 1st and R 2nd) = 1/6 * 2/5 = 1/15. P(Y 1st and B 2nd) = 1/6 * 3/5 = 1/10

(4.3#3)For each of the following situations, explain why the combinations rule or the permutations rule should be used. (a)Determine the number of different groups of 5 items that can be selected from 12 distinct items. (b)Determine the number of different arrangements of 5 items that can be selected from 12 distinct items.

(a) Use the combinations rule, since only the items in the group and not their arrangement is of concern. (b) Use the permutations rule, since the number of arrangements within each group is of interest.

(4.2#17)You roll two fair dice, a green one and a red one. (a)Are the outcomes on the dice independent? (b)Find P(5 on green die and 3 on red die). (c)Find P(3 on green die and 5 on red die). (d)Find P[(5 on green die and 3 on red die) or (3 on green die and 5 on red die)]

(a) Yes. (b) 0.028 (c) 0.028 (d) 0.056

(4.3#19) Compute C7,7

1

(4.3#17) Compute C5,2

10

(4.3#13) Compute P5,2

20

(4.3#25)There are 15 qualified applicants for 5 trainee positions in a fast-food management program. How many different groups of trainees can be selected?

3003

(4.3#9)Four wires (red, green, blue, and yellow) need to be attached to a circuit board. A robotic device will attach the wires. The wires can be attached in any order, and the production manager wishes to determine which order would be fastest for the robot to use. Use the multi-plication rule of counting to determine the number of possible sequences of assembly that must be tested. Hint: There are four choices for the first wire, three for the second, two for the third, and only one for the fourth.

4 * 3 * 2 * 1 = 24 sequences

(4.3#11)Barbara is a research biologist for Green Carpet Lawns. She is studying the effects of fertilizer type, temperature at time of application, and water treatment after application. She has four fertilizer types, three temperature zones, and three water treatments to test. Determine the number of different lawn plots she needs in order to test each fertilizer type, temperature range, and water treatment configuration.

4 * 3 * 3 = 36.

(4.3#15) Compute P7,7

5040

(4.3#23)The University of Montana ski team has five entrants in a men's downhill ski event. The coach would like the first, second, and third places to go to the team members. In how many ways can the five team entrants achieve first, second, and third places?

60

(4.1#7)A recent Harris Poll survey of 1010 U.S. adults selected at random showed that 627 consider the occupation of firefighter to have very great prestige. Estimate the prob-ability (to the nearest hundredth) that a U.S. adult selected at random thinks the occupation of firefighter has very great prestige.

627/1010 =0.62

(4.1#9)An investment opportunity boasts that the chance of doubling your money in 3 years is 95%. However, when you research the details of the investment, you estimate that there is a 3% chance that you could lose the entire investment. Based on this information, are you certain to make money on this investment? Are there risks in this investment opportunity?

Although the probability is high that you will make money, it is not completely certain that you will. In fact, there is a small chance that you could lose your entire investment. If you can afford to lose all of the investment, it might be worthwhile to invest, because there is a high chance of doubling your money.

(4.1#1)List three methods of assigning probabilities.

Equally likely outcomes, relative frequency, intuition.

(4.1#5)A Harris Poll indicated that of those adults who drive and have a cell phone, the probability that a driver between the ages of 18 and 24 sends or reads text messages is 0.51. Can this probability be applied to alldrivers with cell phones? Explain.

No, the probability was stated for drivers in the age range from 18 to 24. We have no information for other age groups. Other age groups may not behave the same way as the 18- to 24-year-olds.

(4.2#1)If two events are mutually exclusive, can they occur concurrently? Explain.

No. By definition, mutually exclusive events cannot occur together.

(4.1#13)On a single toss of a fair coin, the probability of heads is 0.5 and the probability of tails is 0.5. If you toss a coin twice and get heads on the first toss, are you guaranteed to get tails on the second toss? Explain.

No. The probability of heads on the second toss is 0.50 regard-less of the outcome on the first toss.

(4.2#9)Lisa is making up questions for a small quiz on probability. She assigns these probabilities: P(A) = 0.3, P(B) = 0.3, P(A and B) = 0.4. What is wrong with these probability assignments?

P(A and B) is the probability that both events A and B occur. It cannot exceed the probability that either event occurs. When the assigned probabilities are used to get P(A | B), the result exceeds 1.

(4.3#21)There are three nursing positions to be filled at Lilly Hospital. Position 1 is the day nursing supervisor; position 2 is the night nurs-ing supervisor; and position 3 is the nursing coordinator position. There are 15 candidates qualified for all three of the positions. Determine the number of different ways the positions can be filled by these applicants.

P15,3= 2730

(4.3#1)What is the main difference between a situation in which the use of the permutations rule is appropriate and one in which the use of the combinations rule is appropriate?

The permutations rule counts the number of different arrangements of r items out of n distinct items, whereas the combinations rule counts only the number of groups of ritems out of n distinct items. The number of permutations is larger than the number of combinations.