Chapter 4
The following contingency table cross-classifies medical school faculty by the characteristics gender and rank. c. Are events G1 and R3 independent? d. For a medical school faculty member, is the event that the person is female independent of the event that the person is an associate professor? Select the correct choice below and fill in the answer box(es) to complete your choice.
c) No. Events G1 and R3 are only independent if (PR3)=(PR3 | G1) d) No, because P(R2)=0.2180.218 and P(R2 | G2)=0.1780.178 and P(R2)≠P(R2 | G2).
Suppose that E and F are two events and that P(E & F)=0.4 and P(E)=0.5. What is P(F|E)?
0.4/0.5 = 0.8
Identify three ways in which the total number of observations of bivariate data can be obtained from the frequencies in a contingency table.
- Summing the frequencies in the cells - Summing the column totals - Summing the row totals
The probability is 0.305 that the gestation period of a woman will exceed 9 months. In 4000 human gestation periods, roughly how many will exceed 9 months?
E(X)=np n=4000 p=0.305 Roughly [1220] nothing human gestation periods will exceed 9 months.
Based on the given information, decide whether or not the two events in question are independent or whether it is not possible to tell. P(A)=0.4 and P(B∣A)= 0.4
It is not possible to tell if the two events are independent.
An ordinary deck of playing cards has 52 cards. There are four suits—spades, hearts, diamonds, and clubs—with 13 cards in each suit. Spades and clubs are black; hearts and diamonds are red. One of these cards is selected at random. Let A denote the event that a black card is chosen. Find the probability that a black card is chosen, and express your answer in probability notation.
The probability that a black card is chosen is [P(A)]=0.50
An experiment has 4 possible outcomes, all equally likely. An event can occur in 2 ways. Find the probability that the event occurs.
The probability that the event occurs is [0.5].
Suppose P(A)=0.8 and P(B|A)=0.35. Find P(A & B).
The probability P(A & B) is 0.28.
Find the probability P(E or F) if E and F are mutually exclusive, P(E)=0.34, and P(F)=0.51.
The probably P(E or F) is [0.85].
Based on the given information, decide whether or not the two events in question are independent or whether it is not possible to tell. P(C)=0.5, P(D)=0.2, and P(C & D)=
The two events are independent because P(C & D)=P(C)•P(D).
Based on the given information, decide whether or not the two events in question are independent or whether it is not possible to tell. P(B)=0.4 and P(B∣A)=0.6
The two events are not independent because P(B∣A)≠P(B).
The following table provides a frequency distribution for the number of rooms in this country's housing units. The frequencies are in thousands. Rooms No. of units 1 551 2 1,462 3 10,911 4 23,400 5 27,957 6 24,619 7 14,618 8+ 17,300 A housing unit is selected at random. Find the following probabilities. a. Find the probability that the housing unit obtained has four rooms. b. Find the probability that the housing unit obtained has more than four rooms. c. Find the probability that the housing unit obtained has one or two rooms. d. Find the probability that the housing unit obtained has fewer than one room. e. Find the probability that the housing unit obtained has one or more rooms.
a) (probability of 4th room)/total 23,335/120,734=0.194 b) p(x>4)=p(5)+p(6)+p(7)+p(8)=84,417 84,417/120,734=0.699 c) p[x=1 or x=2] = p[x=1] + p[x=2] = 551/120,734 + 1462/120,734 = 0.0048950 + 0.012009 =0.0169~0.017 d) 0 e) 1.00
An ordinary deck of playing cards has 52 cards. There are four suits—spades, hearts, diamonds, and clubs—with 13 cards in each suit. Spades and clubs are black; hearts and diamonds are red. If one of these cards is selected at random, what is the probability that it is a) a heart? b) black? c) not a club?
a) 0.25 b) 0.5 c) 0.75
When one die is rolled, the following six outcomes are possible. A = event the die comes up even B = event the die comes up 5 or more C = event the die comes up at most 2 List the outcomes that constitute each of the following events. a. (not A) b. (A & B) c. (B or C)
a) 1,3,5 b) 6 c) 1,2,5,6
A person has agreed to participate in an extrasensory perception (ESP) experiment. He is asked to randomly pick two numbers between 0 and 9, inclusive. The second number must be different from the first. Let H=event the first number is 3, and K=event the second number picked exceeds 6. a. Determine P(H). b. Determine P(K | H). c. Determine P(H & K). d. Find the probability that both numbers picked are less than 7. e. Find the probability that both numbers picked are greater than 7.
a) 1/10 = 0.1 b) (3/9)= 0.333 c) (1/10)(6/9) = 0.067 d) (7/10)(6/9)=0.467 e) (2/10)(1/9)=0.022
The accompanying figure shows the 36 equally likely outcomes when two balanced dice are rolled. Complete parts (a) through (d) below. a. Determine the probability that the sum of the dice is 10. b. Determine the probability that the sum of the dice is even. c. Determine the probability that the sum of the dice is 7 or 6. d. Determine the probability that the sum of the dice is 5, 8,or 4.
a) 3 outcomes - [6,4] [4,6] [5,5] 3/36=0.083 b) 0.5 c) [6/36] + [5/36] = 11/36 = 0.306 d) [4/36] + [5/36] + [3/36] = 12/36 = 0.333
According to a survey, the distribution of graduate science students in doctorate-granting institutions is shown to the right. Frequencies are in thousands. A graduate science student who is attending a doctorate-granting institution is selected at random. Complete parts a through c. Field Frequency Physical sciences 35.1 Environmental sciences 11.8 Mathematical sciences 18.4 Computer sciences 44.4 Agricultural sciences 12.3 Biological sciences 63.2 Psychology 46.5 Social sciences 87.3 a. Determine the probability that the field of the student obtained is psychology. b. Determine the probability that the field of the student obtained is physical or social science. c. Determine the probability that the field of the student obtained is not computer science.
a) 46.5/319=0.146 b) (35.1/319) + (87.3/319) (0.11)+(0.274)=0.384 c) 274.6/319=0.861
Consider the set consisting of the first 10 positive whole numbers (that is, 1-10). Complete parts (a) through (c) below. a. Determine explicitly the numbers in the set that are at least 5. b. Determine explicitly the numbers in the set that are at most 6. c. Determine explicitly the numbers in the set that are between 4 and 8, inclusive.
a) 5,6,7,8,9,10 b) 1,2,3,4,5,6 c) 4,5,6,7,8
When one die is rolled, six outcomes are possible. These outcomes are those where the die shows each value between 1 and 6. Complete parts (a) through (d). A = event the die comes up even B = event the die comes up 5 or more C = event the die comes up at most 4 D = event the die comes up a. Are events A and B mutually exclusive? b. Are events B and C mutually exclusive? c. Are events A, C, and D mutually exclusive? d. Are there three mutually exclusive events among A, B, C, and D? e. Are all four of the events mutually exclusive?
a) No. At least the outcome where the die comes up 6 is common for these events b) Yes. These events do not share any common outcomes. c) No. At least the outcome where the die comes up 2 is common for A and C. d) Yes. B,C, and D are mutually exclusive e) No. Events A and C are not mutually exclusive because they share at least one common outcome.
The age distribution for politicians in a certain country is shown in the accompanying table. Suppose that a politician is selected at random. Let events A, B, C, and S be defined as follows. Complete parts (a) through (d) below. A=event the politician is under 50 B=event the politician is in his or her 50s C=event the politician is in his or her 60s S=event the politician is under 70 Age (yr) No. of politicians Under 50 11 50-59 29 60-69 37 70-79 19 80 and over 3 a. Use the table and the f/N rule to find P(S). b. Express event S in terms of events A, B, and C. c. Determine P(A), P(B), and P(C). d. Compute P(S), using the special addition rule and your answers from parts (b) and (c). Compare your answer with that in part (a).
a) P(S)=0.778 b) S=(A or B or C) c) P(A)=0.111 P(B)=0.293 P(C)=0.374 d) (0.111)+(0.293)+(0.374)=0.l778 This is [roughly or exactly equal to] the answer in part (a)
A committee consists of five executives, three women and two men. Their names are Maria (M), Robert (R), Elizabeth (E), Gary (G), and Penelope (P). The committee needs to select a chairperson and a secretary. It decides to make the selection randomly by drawing straws. The person getting the longest straw will be appointed chairperson, and the one getting the shortest straw will be appointed secretary. The possible outcomes are shown in the accompanying table. Here, for example, ME represents the outcome that Maria (M) is appointed chairperson and Elizabeth (E) is appointed secretary. List the outcomes constituting each of the four events below. ME EM PM RM GM MP EP PE RE GE MR ER PR RP GP MR ER PR RP GP MG EG PG RG GR a. Event E1 is defined as the event a male is appointed chairperson. What outcomes constitute event E1? b. Event E2 is defined as the event Gary (G) is appointed chairperson. What outcomes constitute event E2? c. Event E3 is defined as the event Penelope (P) is appointed secretary. What outcomes constitute event E3? d. Event E4 is defined as the event one male and one female are appointed. What outcomes constitute event E4?
a) RM, RE, RP, RG, GM, GE, GP, GR b) GM, GE, GP, GR c) MP, EP, RP, GP d) MR, MG, ER, EG, PR, PG, RM, RE, GM, GE, RP, GP
The contingency table provides a cross classification of the weights and years of experience for players on a professional team. Years of Experience Rookie 1-5 6-10 10+ Total Under 200lb [2] [8] [3] [2] [15] 200-300lb [7] [16] [14] [4] [41] over 300lb [4] [8] [0] [0] [12] Total [13] [32] [17] [6] [68] a. How many cells are in this contingency table? b. How many players are on the roster? c. How many players have 10+ years experience? d. How many players weigh under 200lb?
a. 12 b. 68 c. 6 d. 2
A group maintains a database of the number, source, and location of oil spills in a country's navigable and territorial waters. The following is a probability distribution for location of oil spill events. Apply the special probability rule to find the percentage of oil spills in the country's navigable and territorial waters that a. occur in A or B. b. occur in D, E or H. c. do not occur in A, B, D, E, or F. Location Probability A 0.003 B 0.038 C 0.234 D 0.027 E 0.003 F 0.369 G 0.141 H 0.162 Other 0.023
a. 4.1% b. 19.2% c. 56%
An incomplete contingency table is provided. Use this table to complete the following. a. Fill in the missing entries in the contingency table. b. Determine P(C1), P(R2), and P(C1 & R2). c. Construct the corresponding joint probability distribution. C1 C2 Total R1 4 [ ] 10 R2 [ ] 6 [ ] Total [ ] [ ] 25
a. C1 C2 Total R1 4 [6] 10 R2 [9] 6 [15] Total [13] [12] 25 b. P(C1) = 13/25 P(R2) = 15/25 P(C1&R2) = 9/25 c. C1 C2 Total R1 [0.16] [0.24] [0.4] R2 [0.36] [0.24] [0.6] Total [0.52] [0.48] [1]
When one die is rolled, the following six outcomes are possible. List the outcomes constituting A = event the die comes up even, B = event the die comes up 5 or more, C = event the die comes up at most 4, D = event the die comes up 1.
A) 2,4,6 B) 5,6 C)1,2,3,4 D)1
Find the probability P(not E) if P(E)=0.24.
The probability P(not E) is [0.76].
