Math 407 Exam 1
How many different seven digit telephone numbers can be formed if the first digit can not be zero?
(0-9) = 10 numbers 9 • 10 • 10 • 10 • 10 • 10 • 10 = 9,000,000
Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each has a probability of 0.02 of failing to detect a plane in its area. What is the probability that it will correctly detect exactly three aircraft before it fails to detect one, if aircraft arrivals are independent single events occurring at different times?
(0.98)^3 (0.02)
A multiple choice examination has 15 questions each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten correctly?
0
Suppose that 30% of applicants for a certain industrial job posses advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. Find the probability that the first applicant with advanced training in programming is found on the fifth interview.
0.07203
Given that we have already balanced a coin ten times and obtained zero heads, what is the probability that we must toss it at least two more times to obtain the first head?
0.5
An assembly operation in a manufacturing plant requires three steps that can be performed in any sequence. How many different ways can the assembly be performed?
3! = 3 • 2 • 1 = 6
A balanced die is tossed 6 times, and the number in the uppermost face is recorded each time. What is the probability that the numbers recorded are 1, 2, 3, 4, 5, and 6 in any order?
720/46656 = 0.0154
If A and B are independent events, show that A and Bc are also independent. Are Ac and Bc independent? Hint: A = (A intersect Bc) U (A intersect B)
Both are independent. Check work. Problem 2.85 Homework 4
If P(A) > 0, P(B) > 0, and P(A) < P(A|B), show that P(B) < P(B|A).
Check answer. Problem 2.81 Homework 4.
Let Y be a discrete random variable with mean m and variance o^2. If a and b are constants, use Theorems 3.3 through 3.6 to prove that a.) E(aY+b) = am+b b.) V(aY+b) = a^2 o^2
Check answer. Problem 3.33 Homework 5
Suppose that Y is a binomial random variable with n>2 trials and success probability p. Use the technique presented in Theorem 3.7 and the fact that E(Y(Y-1)(Y-2)) = E(Y^3) - 3E(Y^2) + 2E(Y) to derive E(Y^3).
Check answer. Problem 3.55 Homework 6
Show that for any integer n >= 1 a.) (n n) = 1 b.) (n 0) = 1 c.) (n r) = (n n-r) d.) Σ i=0, ∞ (n i) = 2^n
Check work. Problem 2.68 from Homework 3
Prove that (n+1 k) = (n k) + (n k-1).
Check work. Problem 2.69 from Homework 3.
Two cards are drawn from a standard 52-card playing deck. What is the probability that the draw will yield an ace and a face card?
Combinations (4•12)/1326 = 0.0362
If we wish to expand (x+y)^8, what is the coefficient of (x^5)(y^3)? What is the coefficient of (x^3)(y^5)?
Combinations 56x^5y^3 56x^3y^5
The manager of a stockroom in a factory has constructed the following probability distribution for the daily demand (number of times used) for a particular tool. y 0 1 2 p(y) 0.1 0.5 0.4 It costs the factory $10 each time a tool is used. Find the mean and variance of the daily cost for the use of the tool.
E(10Y) = 13 V(10Y) = 41
A particular sale involved four items randomly selected from a large lot that is known to contain 10% defectives. Let Y denote the number of defectives among the four sold. The purchaser of the items will return the defectives for repair, and the repair cost is given by C = 3Y^2 + Y + 2. Find the expected repair cost..
E(C) = 3.96
An oil exploration firm is formed with enough capital to finance ten explorations. The probability of a particular exploration being successful is 0.1. Assume the colorations are independent. Find the mean and variance of the number of successful explorations.
E(Y) = 1 V(Y) = 0.9
Suppose that 30% of applicants for a certain industrial job posses advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. What is the expected number of applicants who need to be interviewed in order to find the first one with advanced training?
E(Y) = 1/p = 3.33
Let Y be a random variable with p(y) given in the table. Find E(Y), E(1/Y), E(Y^2 -1), and V(Y). y 1 2 3 4 p(y) 0.4 0.3 0.2 0.1
E(Y) = 2 E(1/Y) = 0.642 E(Y^2 -1) = 4 V(Y) = 1
A single fair die is tossed once. Let Y be the number facing up. Find the expected value and variance of Y.
E(Y) = 3.5 V(Y) = 35/12
A single fair die is rolled once. Let Y be the number rolled. Find the expected value and variance of Y.
E(Y) = 3.5 V(Y) = 35/12 = 2.92
If A and B are events and B is a subset of , use the results derived in exercise 2.5b and the axioms in definition 2.6 to prove that P(A) = P(B) + P(A intersect Bc)
Let A and B be simple events. Let x ∈ P(B) or P(A intersect Bc). If x ∈ P(B), and B is a subset of A, then x ∈ P(A). If x ∈ P(A intersect Bc), then x ∈ P(A) and x ∈ P(Bc). In both cases, x ∈ P(A). Therefore, P(B) + P(A intersect Bc) is a subset of P(A). [Prove the reverse]. This proves that P(A) = P(B) + P(A intersect Bc).
If A and B are mutually exclusive events, and P(B) > 0, show that P(A) P(A|A U B) = ————————- P(A) + P(B)
Mutually exclusive means that P(A intersect B) = 0. Start from the left side and continue to simplify Problem 2.83 Homework 4
Suppose that A and B are events with P(A) = 0.85 and P(B) = 0.4. Is ur possible for P(A intersect B) = 0.15? What is the smallest possible value for P(A intersect B)? What is the largest that P(Ac U Bc) can be?
P(A intersect B) can not be equal to 0.15. Smallest: 0.25 Largest: 0.75
You and a friend play a game where you each toss a balanced coin. If the upper faces on the coins are both tails, you win $1; if the faces are both heads, you win $2; if the coins do not match, you miss $1 (win -$1). Give the probability distribution for your winnings, Y, on a single play of this game.
P(Y=1) = 0.25 P(Y=2) = 0.25 P(Y=-1) = 0.5
A personnel director for a corporation has hired ten new engineers. If three (distinctly different) positions are open at a Cleveland plant, in how many ways can she fill the positions?
Permutation 10 3 10!/7! = 10 • 9 • 8 = 720
You roll a balanced die once. Using the classical probability model, what is the probability that you roll an even number given that you have rolled a number that is greater than 2? Are these events independent? Why or why not?
Probability: 0.5 Yes, they're independent because P(A|B)=P(A)
You draw five cards from a standard 52 card deck. Assuming you are equally likely to draw any given card, what is the probability of drawing 4 of a kind?
Quiz 2 Question 3
You toss a fair/balanced coin three times and observe the face of the coin each time. Determine the sample space S for this experiment. Assuming the classical probability model, what is the probability that you ross at least two heads?
S = {H, T} x {H, T} x {H,T} = 8 elements Let A denote the event that you toss at least two heads P(A) = 4/8 = 0.5
Every person's blood type is A, B, AB, or O. In addition, each individual either has the Rhesus (Rh) factor (+) or does not (-). A medical technician records a person's blood type and Rh factor. List the sample space for this experiment.
S={A+, B+, AB+, O+, A-, B-, AB-, O-}
Suppose that A and B are independent events such that the probability that neither occurs is a and the probability of B is b. Show that 1-b-a P(A) = ——————- 1-b
Since A and B are independent, use the property that P(A intersect B) = P(A)P(B). Check work. Problem 2.99 Homework 4.
Suppose that A is a subset of B and that P(A) > 0 and P(B) > 0. Are A and B independent? Prove your answer.
Since A is a subset of B, A intersect B = A, so this would mean P(A intersect B) = P(A) = P(A)P(B). But P(A) > 0 or P(B) = 1. But P(B) < 1, thus A and B can not be independent.
A particular concentration of a chemical found in polluted water has been found to be lethal to 20% of fish that are exposed to the concentration for 24 hours. Twenty fish are placed in a tank containing the concentration of chemical in water. a.) Find the probability that exactly 14 survive b.) Find the probability that at least 10 survive c.) Find the probability that at most 16 survive d.) Find the mean and variance of the number that survive
a.) 0.109 b.) 0.9979 c.) d.) E(Y) = 16 V(Y) = 3.2
The probability that a patient recovers from a stomach disease is 0.8. Suppose 20 people are known to have contracted this disease. What is the probability that a.) exactly 14 recover? b.) at least 10 recover? c.) at least 14 but not more than 18 recover? d.) at most 16 recover?
a.) 0.109 b.) 0.999 c.) 0.844 d.) 0.589
In a gambling game, a person draws a single card from an ordinary 52-card deck. A person is paid $15 for drawing a jack or queen, and $5 for drawing a king or an ace. A person who draws any other card must pay $4. If a person plays this game, what is the expected gain?
Y = 15, 5, -4 P(Y) = 8/52, 8/52, 36/52 E(Y) = 0.308
A new surgical procedure is successful with a probability of p. Assume that the operation is performed 5 times and the results are independent of one another. What is the probability that a.) all five operations are successful if p=0.8 b.) exactly four are successful if p=0.6? c.) less than two are successful if p=0.3?
a.) 0.328 b.) 0.2592 c.) 0.5282
Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each has a probability of 0.02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a.) goes undetected? b.) is detected by all three?
a.) (0.02)^3 b.) (0.98)^3
You have in your possession a fair/balanced die. a.) Suppose that you roll the die n times. Assuming the classical probability model, what is the probability that none of the first n-1 tosses are a 6, but the nth toss is a 6? b.) Suppose that you cast the die repeatedly. Show that the probability of eventually rolling a 6 is 1.
a.) (1/5)(5/6)^n b.) U n=1, ∞ En = (1/5) Σ n=1, ∞ (5/6)^n = (1/5) • (5/6)/(1-(5/6)) = (1/5) • (5/6) • (6) = 1
Cards are dealt, one at a time, from a standard 52-card deck. a.) If the first 2 cards are both spades, what is the probability that the next 3 cards are also spades? b.) If the first 3 cards are spades, what's the probability that the next 2 cards are also spades? c.) If the first 4 cards are spades, what's the probability that the next card is also a spade?
a.) 0.00842 b.) 0.03827 c.) 0.1875
A study of the posttreatment behavior of a large number of drug abusers suggests that the likelihood of conviction with a two year period after treatment may depend upon the offender's education. The proportions of the total number of cases falling in four education conviction categories are shown: Education Con. Not Con. Total 10 years or more 0.10 0.30 0.40 9 years or less 0.27 0.33 0.60 Total 0.37 0.63 1.00 A: the offender has 10 or more years of education B: the officer is convicted within two years of completing treatment Find the following: a.) P(A) b.) P(B) c.) P(A intersect B) d.) P(A U B) e.) P(Ac) f.) P((A U B)c) g.) P((A intersect B)c) h.) P(A|B) i.) P(B|A)
a.) 0.40 b.) 0.37 c.) 0.10 d.) 0.67 e.) 0.60 f.) 0.33 g.) 0.90 h.) 10/37 i.) 0.25
If two events, A and B, are such that P(A) = 0.5, P(B) = 0.3, and P(A intersect B) = 0.1, find the following: a.) P(A|B) b.) P(B|A) c.) P(A|A U B) d.) P(A| A intersect B) e.) P(A intersect B|A U B)
a.) 1/3 b.) 1/5 c.) 5/7 d.) 1 e.) 1/7
Suppose that a small country issues a unique license plate for each registered vehicle in the country. The plate consists of 5 characters, consisting of digits from 0 to 9 and letters from A to Z. Assuming that there are no restrictions on plate characters and that any registered vehicle is equally likely to be issued a plate with any combination of characters, find the probability that a.) a randomly selected vehicle has a plate which consists entirely of digits 0 to 9. b.) a randomly selected vehicle has a plate where the first two characters are either an A, a 5, or a 8 and the last two characters are a G and a Z.
a.) 100,000/60466176 = 0.001654 b.) 324/60466176 = 0.000005358
Five cards are dealt from a standard 52-card deck. What is the probability that we draw a.) 1 ace, 1 two, 1 three, 1 four, and 1 five? b.) any straight?
a.) 1024/2598960 = 0.000394 b.) 100,000/2598960 = 0.03848
An experiment consists of tossing a pair of dice. a.) Use the combinatorial theorem to determine the number of sample points in sample space S. b.) Find the probability that the sum of the numbers appearing on the dice is equal to 7.
a.) 21 b.) 3/21
Suppose that an engineering firm has three vacant job postings. After reviewing initial applications, the firm has narrowed down the applicants to 20 people. a.) If each job is the same (with the same pay, benefits, etc.), how many possible ways are there to fill the three postings? b.) If each job is different (each having different pay, benefits, etc.), how many ways are there to fill the three postings?
a.) Combination 1140 b.) Permutation 6840
Suppose that A and B are two events such that P(A) = 0.8 and P(B) = 0.7. a.) Is it possible that P(A intersect B) = 0.1? Why or why not? b.) What is the smallest value for P(A intersect B)? c.) Is it possible that P(A intersect B) = 0.77? Why or why not? d.) What is the largest value of P(A intersect B)?
a.) No b.) 0.5 c.) No d.) 0.7
Diseases I and II are prevalent among young people in certain populations. It is assumed that 10% of the population will contract Disease I sometime in their lifetime, 15% will contract Disease II eventually, and 3% will contract both. a.) Find the probability that a randomly chosen person from this population will contract at least one disease. b.) Find the conditional probability that a randomly chosen person from this population will contract both diseases, given that he or she has contracted at least one disease.
a.) P(A U B) = 0.22 b.) P(A intersect B|A U B) = 3/22
A smoke detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by A is 0.95, by device B is 0.90, and by both 0.88. a.) If smoke is present, find the probability that the smoke will be detected by either device A or device B or both devices. b.) Find the probability that the smoke will be undetected.
a.) P(A U B) = 0.97 b.) P((A U B)c) = 0.03
An oil prospecting firm hits oil or gas on 10% of its drillings. If the firm drills two wells, the four possible simple events and three of their associated probabilities are as given in the accompanying table. Find the probability that the company will hit oil or gas a.) on the first drilling and miss the second b.) on at least one of the two drillings E1 Hit (oil or gas) Hit (oil or gas) 0.01 E2 Hit Miss ? E3 Miss Hit 0.09 E4 Miss Miss 0.81
a.) P(E2) = 0.09 b.) P(E1, E2, E3) = 0.19
A sample space consists of five simple events, E1, E2, E3, E4, and E5. a.) If P(E1) = P(E2) = 0.15, P(E3) = 0.4, and P(E4) = 2P(E5), find the probabilities of E4 and E5. b.) If P(E1) = 3P(E2) = 0.3, find the probabilities of the remaining simple events if you know that the remaining simple events are equally probable.
a.) P(E4) = 0.20, P(E5) = 0.10 b.) P(E3) = P(E4) = P(E5) = 0.20
Five balls numbered 1-5 are placed in an urn. Two balls are randomly selected from the five, and their numbers are noted. Find the probability distribution for the following: a.) the largest of the two sampled numbers b.) the sum of the two sampled numbers
a.) P(Y=2)=1/10, P(Y=3)=2/10, P(Y=4)=3/10, P(Y=5)=4/10 b.) P(Y=3) = P(Y=4) = P(Y=8) = P(Y=9) = 1/10 P(Y=5) = P(Y=6) = P(Y=7) = 2/10
Patients arriving at a hospital outpatient clinic can select one of three stations for service. Suppose that physicians are assigned randomly to the stations and that the patients therefore have no station preference. Three patients arrive at the clinic and their selection of stations is observed. a.) List the sample points for the experiment. b.) Let A be the event that each station receives a patient. List the sample points in A. c.) Make a reasonable assignment of probabilities to the sample points and find P(A).
a.) S = {1,2,3} x {1,2,3} x {1,2,3} = 27 elements b.) A = {(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1)} c.) P(A) = 6/27 = 2/9
The proportions of blood phenotypes, A, B, AB, and O, in the population of all Caucasians in the United States are approximately 0.41, 0.10, 0.04, and 0.45, respectively. A single Caucasian is chosen at random from the population. a.) List the sample space for this experiment. b.) Make use of the information given to assign probabilities to each of the simple events. c.) What is the probability that the person chosen at random has either A or AB type blood?
a.) S = {A, B, AB, O} b.) P(A) = 0.41, P(B) = 0.10, P(AB) = 0.04, P(O) = 0.45 c.) P(A, AB) = 0.45
Suppose two balanced coins are tossed and the upper faces are observed. a.) List the sample points for this experiment b.) Assign a reasonable probability to each sample point. c.) Let A denote the event that exactly one head is observed and B the event that at least one head is observed. List the sample points in both A and B. d.) From your answer on part c, find P(A), P(B), P(A intersect B), P(A U B), and P(Ac U B).
a.) S = {H, T} x {H, T} = {(H, H), (H, T), (T,H), (T,T)} b.) P(H,H) = P(H, T) = P(T,H) = P(T,T) = 0.25 c.) A = {(H, T), (T,H)} B = {(H,H), (H,T), (T,H)} d.) P(A) = 0.5 P(B) = 0.75 P(A intersect B) = 0.5 P(A U B) = 0.75 P(Ac U B) = 1
