Homework 2 Sample Problems and Solution
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North Carolina State University *
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724
Subject
Mathematics
Date
Jun 19, 2024
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14
Uploaded by rogerzvc
ISEN 370 Homework 2 Sample Questions and Solutions 1. List the elements of each of the following sample spaces. Complete parts (a) through (e) below. (a) The set of integers between 1 and 50 divisible by 6. The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol S. The sample space for this set is all integers that satisfy the given condition. The set of integers between 1 and 50 that are divisible by 6 is the same as the multiples of 6. The set that represents the multiples of 6 between 1 and 50 is S={6,12,18,24,30,36,42,48}. (b) The set 𝑆𝑆
= {
𝑥𝑥
|
𝑥𝑥
2
+ 2
𝑥𝑥 −
15 = 0}
. 𝑆𝑆
= {
𝑥𝑥
|
𝑥𝑥
2
+ 2
𝑥𝑥 −
15 = 0}
. Sample spaces with a large or infinite number of sample points are often described by a statement or rule method. The vertical bar in the rule is read “such that”. For this set, the rule is 𝑥𝑥
2
+ 2
𝑥𝑥 −
15 = 0
Solve the equation for x (x+5)(x-3)=0 x = -5,3 Thus, the set 𝑆𝑆
= {
𝑥𝑥
|
𝑥𝑥
2
+ 2
𝑥𝑥 −
15 = 0}
contains the elements -5 and 3. (c) The set of outcomes when a coin is tossed until a tail or two heads appear. Let H represent a head and T represent a tail. List the possible outcomes of the first two tosses. HT, HH, T List all of the outcomes for the set. S={HT,HH,T} (d) The set S= {x | x is a continent}
For this set, the rule is that x must be a continent. List the continents. S = {N. America, S. America, Europe, Asia, Africa, Australia, Antarctica} (e) The set 𝑆𝑆
= {
𝑥𝑥
|4
𝑥𝑥 −
20
≥
0 𝑎𝑎𝑎𝑎𝑎𝑎
𝑥𝑥
< 4}
. The rule for this set is 4
𝑥𝑥 −
20
≥
0 𝑎𝑎𝑎𝑎𝑎𝑎
𝑥𝑥
< 4
Solve the inequality 4x
−
20
≥
0 for x. 4x
−
20 ≥
0 x ≥
5 Therefore, the rule for this set is x
≥
5 and x<3. There are no real numbers that satisfy both of these inequalities. Therefore, the set S is the empty set. S=Φ
2. An experiment involves tossing a pair of dice, one green and one red, and recording the numbers that come up. If x equals the outcome on the green die and y equals the outcome on the red die, describe the sample space S. Complete parts (a) and (b) below. (a) Describe the sample space S by listing the elements (x,y). X: 1,2,3,4,5,6 Y: 1,2,3,4,5,6 The elements (x,y) will have 36 combinations as shown below: S {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} (b) Describe the sample space S by using the rule method X takes value between 1 and 6, y takes value between 1 and 6. S= { (x,y) | 1 ≤ x ≤ 6 and 1 ≤ y ≤ 6}
3. An experiment consists of tossing a die and then flipping a coin twice if the number on the die is 4 or greater. If the number on the die is 3 or less, the coin is flipped once. Using the notation
H, for example, to denote the outcome that the die comes up and then the coin comes up heads, and HT to denote the outcome that the die comes up followed by a head and then a tail on the coin, construct a tree diagram to show the 18 elements of the sample space S. Tree diagram is shown below: 4. Two jurors are selected from 4 alternates to serve at a murder trial. Using the notation A1A3, for example, to denote the simple event that alternates 1 and 3 are selected, list the 6 elements of the sample space S. Answer: The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol S. Each outcome in a sample space is called an element or a member of the sample space, or simply a sample point. If the sample space has a finite number of elements, list the members separated by commas and enclosed in braces. The outcome A1A2 is possible since it represents the event that alternates 1, and 2 are selected. The outcomes A1A2 and A2A1 shouldn't both be included in the sample space because the order in which alternates are selected is not significant. The outcome A1A1shouldn't be included in the sample space because an alternate cannot be repeatedly selected.
Since there are a total of 6 outcomes in the sample space, list all the outcomes separated by commas and enclosed in braces. S={A1A2, A1A3, A1A4, A2A3, A2A4, A3A4} 5. An experiment consists of tossing a die and then flipping a coin twice if the number on the die is 4 or greater. If the number on the die is 3 or less, the coin is flipped once. the notation 4H, for example, to denote the outcome that the die comes up 4 and then the coin comes up heads, and 3HT to denote the outcome that the die comes up 3 followed by a head and then a tail on the coin. Complete parts (a) through (e) below. Develop a tree diagram (a) List the elements corresponding to the event A that a number greater than 4 occurs on the die. A={5HH,5HT,5TH,5TT, 6HH, 6HT,6TH,6TT} (b) List the elements corresponding to the event B that a head then a tail occurs B={4HT,5HT,6HT} (c) List the elements corresponding to the set A′
A’={1H,1T,2H,2T,3H,3T,4HH,4HT,4TH,4TT} (d) List the elements corresponding to the set B. A′ ∩B
A′ ∩B
={4HT}
(e) List the elements corresponding to the set A
Ս
B. A
Ս
B={4HT, 5HH,5HT,5TH,5TT, 6HH, 6HT,6TH,6TT } 6. An engineering firm is hired to determine if certain waterways are safe for fishing. Samples are taken from two rivers. Complete parts (a) through (c) below. (a)
List the elements of a sample space S, using the letters F for safe to fish and N for not safe to fish. S={FF, FN,NF, NN} (b) List the elements of S corresponding to event E that at least one of the rivers are safe for fishing. E={FF,FN,NF} (c) Define an event that has as its elements the points {FF, NF} The second river was safe for fishing. 7. Construct a Venn diagram to illustrate the possible intersections and unions for the following events relative to the sample space consisting of all automobiles made in the United States. B: Blind spot warning P: Power steering R: Reverse camera 8. If S={0,1,2,3,4,5,6,7,8,9} and A={1,3,5,7,9}, B={0,2,4,6,8}, C={2,3,4,5} and D={0,7,8}, list the elements of the sets corresponding to the following events: (a) List the elements of the set corresponding to A
ꓴ
C. (b) List the elements of the set corresponding to A
ꓵ
B. (c) List the elements of the set corresponding to C’.
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