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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

A manufacturing item exhibits two types of faults, say ???? and ????, which occur independently of one another. The probability of a fault, which is type-????, is 12%, while a type-???? fault has a
If an event A is independent of an event B, and B is independent of another event C, is it always true that A and C are independent? Prove or disprove (using a counterexample in the latter case) this
If for three independent events A, B, and C of a sample space, we know that P(AB) = 0.3, P(AC) = 0.48, P(BC) = 0.1, find the probability of the event A ∪ B ∪ C.
Let A and B be two events on a sample space. Then, show the following:(i) If P(A) = 0, this implies that P(AB) = 0.(ii) If P(A) = 1, then we have P(A ∪ B) = 1; use this to establish that P(AB) =
If the events A and B are independent and A ⊆ B, show that P(A) = 0 or P(B) = 1.
If A and B are two events on a sample space such that 0 < P(A), P(B) < 1, verify that each of the following conditions is equivalent to the independence of A and B:(i) P(A|B) + P(A′) = 1;(ii)
Pat takes part in a quiz show with multiple choice questions. There are three possible answers to each question. The probability that she knows the answer to a question is 80%. If Pat does not know
We have n chips numbered 1, 2,…, n. Tom, who likes fancy experiments, selects a chip at random and if the number on that chip is i, he tosses a coin i times(1 ≤ i ≤ n). Tom has just completed
Among male smokers, the lifetime risk of developing lung cancer is 17%; among female smokers, this risk is 12%. For nonsmokers, this risk is significantly lower:1.3% for men and 1.5% for women.
A telecommunications system transmits binary signals (0 or 1). The system includes a transmitter that emits the signals and a receiver which receives those signals. The probability that the receiver
Diana is about to go out with her friends and her mother asks her how much money she has in her purse. Diana says she has either a $10 note or a $20 note, but she can’t remember. Her mother puts in
In a certain company, there are three secretaries responsible for typing the mail of the manager. When she types a letter, Secretary A has a probability of 0.04 for making at least one misprint,
Suppose that in a painting exhibition, 96% of the exhibits are genuine, while the remaining 4% are fake. A painting collector can identify a genuine painting as such with a probability 90%, while if
Electric bulbs manufactured in a production unit are packaged in boxes, with each box containing 120 bulbs. The probability that a box has i defective bulbs is 1∕5, for each i = 0, 1, 2,…, 4. If
A box B1 contains four red and six black balls, while a second box B2 contains seven red and three black balls. We select a ball from B1 and place it in B2. Then, we pick up a ball from B2 at random
A box B1 contains 3 red and 6 blue balls, a second box B2 contains 7 red and 7 blue balls, while a third box B3 has 5 red and 9 blue balls. We select a box at random and then from this box we pick a
A motor insurance company classifies its customers as good drivers (G) and bad drivers (B). 65% of the company’s customers are classified as G. The probability that a good customer makes a claim in
Students at a University take a Probability exam in three classrooms. The number of students who are well-prepared (W) and poorly-prepared (P) for the exam in each of the three classrooms are as
Assume that in a lottery, 20 balls numbered 1 to 20 are put in a large bowl and then 3 balls are selected, one after the other, at random and without replacement. What is the probability that the
We throw a die and if the outcome is k (1 ≤ k ≤ 6), then we select a ball from an urn that contains 2k white balls and 14 − 2k black ones. Show that the probability of selecting a white ball is
We throw a die and, if the outcome is i, then we toss a coin i times. What is the probability that in these coin tosses,(a) no heads appear?(b) only one face of the coin appears, that is if we toss
John has a red and a blue die and throws them simultaneously.(i) What is the probability that the outcome of the blue die is larger than that of the red die?(ii) Find the probability that the
A factory has three production lines that produce 50%, 30%, and 20%, respectively, of the items made in the factory during a day. It has been found that 0.7% of the items produced in the first line
Among the drivers insured with an insurance company, 45% made no claims during a year, 35% made one claim, and 20% made at least two claims. The probabilities that a driver will make more than one
From an usual pack of 52 cards, we select a card at random. Then we select another card from the remaining 51 cards. What is the probability that the second card chosen is(a) an ace?(b) a diamond?(c)
A bowl contains six white and five red balls, while a second bowl contains three white and seven red balls. We select randomly a ball from the first bowl and place it in the second. Then, we choose
Sixty percent of the students in a University class are females. If, among the female students, 25% have joined the University Sports Club to do at least one sport, and the corresponding percentage
A University degree program enrolled this year r female and s male students. If students are registered at the University in a completely random order, what is the probability that, for k ≤ min(r,
An urn contains a red and b green balls.We select k balls without replacement with k ≤ min{a, b}. Show that the probability all selected balls are of the same color equals(a)k + (b)k(a +
In an oral exam at a University, the course lecturer has to examine r female and s male students. The order in which the students are examined is assumed to be random.Consider the events A: all
A pharmaceutical company produces boxes of tablets for a particular disease. Each box contains 20 tablets. The quality control unit of the company selects a box at random and examines the tablets to
An insurance company classifies the claims arriving as being either low (L) or high(H). On a certain day, 21 claims arrived, 12 of which were L. At the end of the day, a company employee registers
During a football season in the English Premier League in football, Manchester United won 25 games, had 9 draws, and lost 4 games. If we do not know the order that United faced their opponents, so
Maria has bought a toy which contains a bag with the 26 letters of the alphabet in it.(i) Maria selects five letters at random. What is the probability that the letters she chose can be rearranged so
Tom has a bowl that contains four white balls and three red balls. He selects balls successively from the bowl (at random and without replacement) and puts them one next to the other.The following
From an usual pack of 52 cards, we select cards without replacement until the first diamond is drawn. What is the probability that this happens with the 4th card drawn?(Hint: Let Ei be the event that
With reference to Example 3.6, suppose that there are m different types of coupons and Jimmy buys r packs of cereals (with r < m).(i) What is the probability that the coupons contained in these r
Kate is in the final year of her studies and she has to choose exactly one of two optional courses offered this semester. She would prefer to take Course I, which she likes best, but she feels that
(The prisoner’s dilemma) Three prisoners, A, B, and C, are sentenced to death and they have been put in separate cells. All three have equally good grounds to apply for parole and the parole board
Suppose A1, A2, and A3 are three events on a sample space Ω and let B be another event such that P(B) > 0. Show that P(A1 ∪ A2 ∪ A3) = S1 − S2 + S3, where S1 = P(A1|B) + P(A2|B) + P(A3|B), S2
Prove Property (f) of Proposition 3.2 directly using the definition of conditional probability (Definition 3.1).
A large PC manufacturing unit has 1000 CPU (central processing units) with speed 2.6 GHz. Each unit has been labeled with a number from 1 to 1000. The same manufacturer has also 1750 CPU with speed
Paul selects 6 cards from a pack of 52 cards and announces that three of them are spades. What is the probability that all six cards selected are spades?
Let A and B be two events in a sample space Ω. Prove that P(A|B) > P(A) holds if and only if P(B|A) > P(B). In such a case, the two events A and B are said to be positively correlated since the
Stephie, who is a theater-lover, attends a theater performance every week in one of the 25 theaters in her city. This year, 11 of these performances are comedies, while the remaining 14 are dramas.
In a large company, there are 500 electronic systems installed. Each of them is either connected to a network (N) or functions as a separate unit (U). Also, some of them have incorporated a new
Let A, B, and C be three events in a sample space Ω. Assuming that the following inequalities hold P(A|B) ≥ P(C|B) and P(A|B′) ≥ P(C|B′), verify that P(A) ≥ P(C).
Mary selects three cards at random from a regular 52-card pack without replacement.Let Ai be the event that the ith card drawn is a Queen for i = 1, 2, 3. Calculate the probabilities(i) P(A2|A1);(ii)
The percentage of unemployed women in a population is 14%, while the general unemployment rate in the population is 11%. Assuming that the two sexes to be equally likely, we select a person at random
Henry throws two dice simultaneously. He observes the outcomes of the two throws and tells us that the two dice showed different faces.What is the probability that the sum of the two outcomes is(i) a
Suppose that Paul selects three cards at random without replacement. Find the probability that the third card drawn is a spade given that the first two cards included k spades. Give your answer for k
Paul selects a card at random from a pack of 52 cards, and then selects a second one among the 51 remaining cards (i.e. without replacement).What is the probability that the second card drawn is an
Andrew tosses three coins. Find the probability that all three coins land heads if we know that(i) the first of the three coins landed heads;(ii) at least one coin landed heads.
Nicky throws a die three times in succession. Consider the events A: the outcome of the second throw is a four;B: two throws out of the three resulted in a four.Calculate the probabilities P(B|A) and
The percentages of people with each of the four blood types (O, A, B, and AB) in Iceland are as follows:type O: 56%; type A: 31%; type B: 11%; type AB: 2%.For a certain person in Iceland, we know
In the examination of a Probability I course at aUniversity, 180 students participated in the exam. Among these students, 80 study for a Mathematics degree, 60 for a Statistics degree, and 40 are in
3.37 Test-Interviews, continued Refer to Exercise 3.36.a. Find the correlation coefficient, r, to describe the relationship between the two tests.b. Would you be willing to use the second and quicker
3.19 LCD TVs, continued Refer to Exercise 3.18. Suppose we assume that the relationship between x and y is linear.a. Find the correlation coefficient, r. What does this value tell you about the
3.11 Refer to Exercise 3.10.a. Use the data entry method in your scientific calculator to enter the six pairs of measurements. Recall the proper memories to find the correlation coefficient, r, the
3.10 A set of bivariate data consists of these measurements on two variables, x and y: (3,6) (5,8) (2,6) (1, 4) (4,7) (4,6)a. Draw a scatterplot to describe the data.b. Does there appear to be a
Find the correlation coefficient for the number of square metres of living area and the selling price of a home for the data in Example 3.5.
How to Calculate the Regression Line
How to Calculate the Correlation Coefficient
A water network has three connections, C1,C2,C3, as shown in Figure 1.18. For each connection, at the places marked 1, 2, 3, some switches have been put and at a particular instant, any switch can be
Let Ω be a sample space and suppose we have defined a set function, P(⋅), which satisfies the properties P1–P3 of Definition 1.10, on that space. Examine whether each of the set functions
Let A1, A2,…, An be an arbitrary collection of n events in a sample space Ω. Show thatThis is known as Bonferroni’s inequality.(Hint: Apply Boole’s inequality from the last exercise to the
We consider the events A1, A2,…, An of a sample space Ω and, from these events, we form n new events B1, B2,…, Bn defined as follows: B1 = A1, while for i = 2, 3,…, n,(i) Verify that the
On a particular day, a restaurant has a special three-course menu with the following choices:Poppy, who is visiting this restaurant with her friends, is to choose one course from each category
For the experiment of throwing a die twice, we consider again the events A, B,C, and D from the last exercise and denote by E the event “exactly two among the events A, B,C, and D occur”:(i)
In the experiment of throwing a die twice, consider the following events:A: the first outcome is 6;B: the second outcome is 4;C: the sum of the two outcomes is 9;D: the first outcome is greater than
Assume that the probability of each elementary event {i} defined in the sample space Ω = {1, 2, 3,…} is given by P({i}) = 5i−1∕7i, i = 1, 2, 3,….Let us define the events An = {n, n + 1, n +
Assume that A, B, and C are three events in a sample space Ω. Show that the following relations hold:(i) P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A′BC) − P(AB′C)−P(ABC′) −
Let A and B be events in a sample space Ω for which it is known that P(A) ≥ a and P(B) ≥b, where a and b are given real numbers.(i) Show that P(AB) ≥ a + b − 1.(ii) If a + b > 1, what do you
The claims arriving at an insurance company are classified by the company as either Large (L) or Small (S) according to their size. The company wants to study the number of claims arriving prior to
Lena, Nick and Tom, who work for the same company, when they arrive for work one morning, meet at the ground floor elevator of the company building, which has three floors above the ground floor.
In a major athletics competition, an athlete has three attempts to clear a certain height. If he succeeds in his first or second attempt, he makes no other attempts on this height.(i) Write down a
When Carol visits the local supermarket she buys a pack of crisps with probability 0.3, a chocolate with probability 0.4 and her favorite fruit juice with probability 0.6. The probability that she
At a large University class, there are 140 male students, 40 of whom own a car, and 160 female students, 20 of whom own a car. Let A be the event that “a randomly selected student is female”
For the disjoint events A and B in a sample space Ω, we know that P(A ∪ B) = 5∕6, 4P(A) + P(B′) = 1.Then, the probabilities of the events A and B are, respectively, equal to(a) P(A) = 1∕6,
We toss a coin successively. Let Bi be the event that the outcome of the ith toss is Heads. Then, the event “Heads occur for the first time at, or after, the second toss”can be written, in terms
We throw a die until a six appears for the first time, at which point the experiment stops. Let Ai be the event that the outcome of the ith throw is a six. The event “a six appears for the first
Let A, B, and C be three events in a sample space Ω, such that A ∪ B ∪ C = A. Then, the following is always true:(a) B ∪ C = A (b) B ∪ C ⊆ A (c) A ⊆ B ∪ C(d) BC = A (e) A ⊆ BC.
Which of the following statements is correct with reference to the sample spacesΩi, for i = 1, 2, 3, 4, 5, defined in the last problem?(a) Ω1 and Ω5 are the only sample spaces which are finite(b)
Maria is waiting at a bus stop for her friend Sarah so that they meet and go to a concert together.Maria wants to know how long she will have to wait until the next bus arrives at the bus stop and
For the events A and B in a sample space Ω, we know that P(A ∪ B) = 1∕6. Then, the value of P(A′) − P(B) is(a) always equal to 5∕6 (b) equal to 5∕6 provided that AB = ∅(c) always equal
In a single throw of a die, consider the events A = {1, 2, 3} and B = {2, 4, 6}. The event (B − A)′ is equal to(a) {4} (b) {2, 4, 5, 6} (c) {1, 3}(d) {4, 6} (e) {1, 2, 3, 5}.
Marc tosses a coin until “Heads” appear for the second time. He is interested in the number of “Tails” which appear before the second appearance of “Heads.” Then, a suitable sample space
Let A, B, and C be three events in a sample space Ω, such that P(A) = 2P(B)and P(B) = 2P(C). If in addition we have A = (B ∪ C)′, then the probability of the event A is(a) 1∕7 (b)2∕7
For the events A and B in a sample space Ω, we know that P(A) = (1 + ????)∕3, P(B) = 1 − ????2 for some real number ????. The admissible range of values for ???? is(a) ???? ≥0 (b) 0 ≤ ????
The event that exactly one of the three events A, B, and C in a sample space Ωoccurs is(a) (ABC)′ (b) (A ∪ B ∪ C)′(c) AB′C′ (d) (AB′C′) ∪ (A′BC′) ∪ (A′B′C)(e) AB ∪ C
Let A1, A2,… be a monotone sequence of events defined in a sample space Ω. If it is known thatthen the probability of the event that at least one of the Ai’s occur is equal to zero. P(A)=() for
Let A1, A2,… be a sequence of events defined in a sample space Ω, and let a new sequence {Cn}n≥1 be defined byThen, {Cn}n≥1 is a decreasing sequence of events in Ω. 00 CA = 1, 2,.... n i=n+1
Let A1, A2,… be a sequence of events defined in a sample space Ω, and let a new sequence {Bn}n≥1 be defined by Bn = A1A2 · · · An, n = 1, 2,….Then, {Bn}n≥1 is a decreasing sequence of
If two events A and B, defined in the same sample space Ω, are mutually exclusive, then A′ ∪ B = Ω.
If for the events A and B we know that P(A) − P(B) = 1∕3, then P(A − B) ≤ 1∕3.
Let A and B be two events in a sample space Ω with A ⊆ B. Then, A′ ∪ B = Ω.
Let A and B be two events in a sample space Ω such that A ⊆ B, and C be another event in Ω. If the events B and C are disjoint, then A and C will also be disjoint events.
If P(A) = 1∕4 and P(A′) = 5P(B) − 1, then the probability of the event B is also 1∕4.
For any events A, B, and C in the same sample space, we have A(BC) = (AB) ∪ (AC).
We toss a coin 300 times and observe that in 160 of these tosses the outcome is“Heads.” Based on this, the relative frequency of the event “Heads occur in a single toss of a coin” is 160.

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