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introduction to probability statistics

Introduction To Probability And Statistics 3rd Edition William Mendenhall - Solutions

At a hospital, 15 babies are born on a given day. If we know that seven of the newborn babies are boys, what is the probability that all seven boys were born in the last seven births at the hospital on that day?
A computer selects digits at random. How many digits are needed to be chosen so that the digit seven appears with a probability of at least 1∕2?
Plates are produced at a certain factory department in dozens. Assume that the probability of a plate being nondefective is p, while the probability of a defective plate is q = 1 − p. Let X be the number of nondefective plates in a dozen, so that X ∼ b(12, p). From past experience, it has been
.(i) In a family with four children, what is the probability of having two boys and two girls, assuming that both genders are equally likely?(ii) If we select 10 families, each having four children, find(a) the probability that at least seven of them will have two boys and two girls;(b) the
Peter plays a game in which he throws a die four times and he wins a dollars if at least one six appears. He is offered an alternative option: to throw two dice 24 times and win a dollars if a double six appears at least once. Should he switch to this new game?
Suppose a mouse moves on a line at random so that, at each step, it either makes a move to the left or to the right with equal probability. Find the probability that the mouse will be at the position where it started after (i) 10 moves; (ii) 20 moves.
If we throw a well-balanced die seven times, what is the probability that(i) no sixes or aces will appear?(ii) at least twice the outcome will be an odd integer?(iii) no more than three times the outcome will be greater than two?
Write down the probability function, the cumulative distribution function, the mean, and variance of the Bernoulli distribution for this experiment.
An urn contains 100 balls, numbered 1–100. We select a ball at random. We consider as success the event that the number on the ball selected is divisible by
If four dice are thrown simultaneously, find the probability that(i) the number six appears in exactly one die;(ii) the number six appears in at least one die.
Jimmy plays another game for which if he wins, he receives ???? dollars. If the probability that he wins the game is 2∕3, the amount he should pay if he loses so that the game is fair equals(a) ???? (b) 2???? (c) 3????∕2 (d)2????∕3 (e)????∕2
Under the assumptions for the lottery in the previous exercise, suppose that Wendy has just bought a lottery ticket.(i) What is the probability that among the six numbers she selected, at least four will appear in the next draw?(ii) Wendy decides to buy a lottery ticket for each of the following
For a family with n children (n ≥ 2), the probability that there exists at least one boy and at least one girl is 1 (a) 1 (b) 1 2"-1 2" 112 (c) (d) (e) 2"-1 2" 2"-1 2" 112
In a lottery with 49 balls numbered 1–49, 6 numbers are selected at any draw. If someone selects six numbers in advance, the probability that all six will be drawn is(49 6)−1.
The minimum number of children a family must have so that it has at least one boy with a probability not less than 90% is four (assume again that it is equally likely for a boy or a girl to be born).
Assuming that both genders are equally likely, the probability that a family with n children has at least one girl is 1 − 2−n.
An urn contains a red and b black balls.We select a ball randomly, note its color, but do not return it to the urn.We repeat this experiment n times (where n ≤ min{a, b})and let X be the number of red balls selected. Then the expected value of X is an∕b.
If it is known that the number of motor accidents in a city during a week has the Poisson distribution with parameter 1, then the probability of having at least one accident during a week is 1 − 2e−1.
For n ≥ 20 and p ≤ 10∕n, the following approximation can be used:(1 − p)n ≈ e−np.
An urn contains a white and b black balls. We select a ball randomly, note its color and return it to the urn. We repeat this experiment and let X be the number of trials until a black ball is selected for the first time. Then, X has a geometric distribution with parameter p = a∕(a + b).
Consider a sequence of Bernoulli trials with the same success probability, p. The expected number of trials until the first failure is equal to 1∕p.
If X has a geometric distribution, then P(X ≤ k + r|X ≤ k) = P(X ≤ r)for any nonnegative integers k and r.
If X has a geometric distribution, then P(X > k + r|X > k) = P(X > r)for any nonnegative integers k and r.
If X has a binomial distribution with parameters n and p, then the random variable Y = n − X also has a binomial distribution with parameters n and p.
Twenty persons of the same age and the same health status are insured with a certain company. From the company’s records, it is estimated that each person of that age has a chance of 0.65 of being alive in 20 years’ time. The expected number of persons that will be alive in 20 years’ time is
Let X be a random variable denoting the number of roulette spins until a black-colored number appears. Then, the distribution of X is geometric. (To answer the question, it is immaterial how many numbers, such as 36, 37, 38, etc., there are on the roulette wheel).
The owner of a restaurant allocates the shifts to the waiters of the restaurant in a way such that each waiter has a probability of 3∕10 of having a day off on a Saturday(where tips are usually higher). If the random variable X denotes the number of weeks a waiter has to wait until he works on a
Make a table similar to the one above, and fill in the corresponding values of the probability functions, for the distributions in Exercise 7.
Fill in the table below with the values of the probability functions for the distributions mentioned in the table. What do you observe? Make a similar table when the values of the probability function are replaced by those of the cumulative distribution function.
The number of characters in a page that are misprinted has the Poisson distribution with parameter
Then, the probability of having at least three misprinted characters on any particular page equals 5e−2.
The number of house burglaries in a city during a day follows the Poisson distribution. If the probability that there will be no burglaries on a particular day is e−5, the expected value of burglaries for a day is 5.
The number, X, of births at a hospital during an hour has the Poisson distribution with parameter ????. We know that the probability a single birth occurs in an hour is four times the probability of having two births in an hour. The value of the parameter ???? is(a) 1∕4 (b)1∕2 (c) 2 (d) e−2
An electric wire of length 30 m has on average 0.5 faults. Assuming that the number of faults follows a Poisson process, the probability that there are exactly two faults in a wire which is 60 m long is(a) 2e−2 (b) e−1 (c) e−1∕2 (d)e−2 (e) e−2∕2
Faye asks George to select a nonnegative integer less than 1000. Assuming that George selects this number completely at random, the probability that this number has at least one digit equal to 2 is(a) 1 − (9∕10)3 (b) 1 − (8∕10)3 (c) 83∕103(d) 93∕103 (e) 1 − (83 + 93)∕10.
The percentage of companies that had their stock price increased at the NY Stock Exchange on a particular day is 45%. If we take a sample of 15 companies, the number of those who did not have an increase on that day has a distribution which is(a) Nb(15, 0.45) (b) (6.75) (c) G(0.45)(d) b(15,
During the last 10 years, 18 fatal accidents have been recorded on a certain motorway. The distribution of the number of accidents in this motorway per year can be reasonably assumed to be(a) Poisson (b) binomial (c) hypergeometric(d) negative binomial (e) none of the above
Apples are packaged at a fruit packaging unit in boxes with 20 fruits each. It is known that 5% of the apples that arrive at the packaging unit cannot be sold, and for this purpose each apple is checked before packaging and put in a box only if found suitable for sale. The distribution of the
We throw a fair die repeatedly until an outcome of either 3 or 4 appears for the fourth time. If X denotes the number of trials until this happens, then which of the following statements is correct?(a) E(X) = 12 (b) E(X) =3 (c)P(X = 4) = (2∕3)4(d) P(X = 3) = (1∕3)4 (e) P(X = 5) = 26∕35
At an industrial unit which produces TFT 22′′ computer screens, the probability that a new screen is defective is 0.05. In a sample of 60 screens, the probability that at most one defective screen is included equals(a) 0.191 55 (b) 0.046 07 (c) 0.145 48 (d) 0.225 88 (e) 0.002 42
The probability that a students fail a certain exam has been estimated to be 0.05.The probability that at least three scripts will be needed until the professor who marks the scripts finds the second script to be a fail is(a) 1 − (0.05)2(0.95) (b) 1 − (0.05)2 (c) (0.05)2 + (0.05)2(0.95)(d)
In a lottery with 49 numbered balls, 6 are selected on any particular draw. If we select 10 numbers before a certain draw, the probability that we have exactly 5 winning numbers is e (19).39 5.10 (b) (c) 49 49 (19)(39) (*) (d) 10 (19). 39+ (10) .39 49 5 (e) (9-19) 49 5
We throw a fair die repeatedly until an outcome of 4 or greater appears for the fifth time. If X denotes the number of trials until this happens, then X follows the(a) Poisson distribution with parameter ???? = 1∕2(b) hypergeometric distribution(c) geometric distribution with parameter p =
Then, the probability that the employee serves exactly four customers in a particular one-hour interval is (a) e-10 104 4 (d) e-10 104 4! 1-(9) (1+1+1+1)( (e) e-10 4! 104
The number of customers served by a bank employee during an hour has the Poisson distribution with mean
A box contains a red and b black balls.We select a ball randomly, note its color but do not return it to the urn.We repeat this experiment n times (where n ≤ min{a, b})and let X be the number of red balls selected. The distribution of X is(a) binomial (b) negative binomial (c) hypergeometric(d)
Claims arrive at an insurance company according to a Poisson process {N(t) ∶t ≥ 0} with rate ???? = 3 claims per day. The probability that there will be exactly one claim in a two-day period is e−6.
When we throw five fair coins, the probability that three or more heads will appear is 0.5.
Again, try to justify your findings.
Draw the graph of the binomial distribution with n = 10, 50, 100, 200, 300, 500, and p = 5∕n. Then compare, in each case, the graph with that of the Poisson distribution with ???? =
Let X ∼ b(n, p) and Y ∼ (????). If ???? = np, so that X and Y have the same mean, consider the sequenceShow that the sequence ????n is first increasing and then decreasing, attaining a maximum at the largest integer not exceeding ????. P(X = n) P(Y = n)' n = 0, 1, 2,...
Consider the probability function of the (????) distributioni) Show that f (x) can be calculated via the recursionwhere a = 0 and b = ????, with initial condition f (0) = e−????.(ii) Verify that f (x) > f (x − 1) if and only if x (iii) Prove that (a) if ???? is not an integer, then f
Let X be a random variable with the Poisson distribution with parameter ????. Show that the rth factorial moment of X,????(r) = E[X(X − 1) · · · (X − r + 1)], r ≥ 1,is given by ????(r) = ????r.
If Y is the variable that represents the number shown in the counter, write down the probability function of Y, and hence find its mean and variance.(The distribution of Y in this exercise is an example of what is called a clumped Poisson distribution.)
The number of particles emitted by a radioactive source in one minute is a random variable following the Poisson distribution with parameter ???? = 2.5. In order to count the number of emissions in one minute, we put a Geiger counter device close to the source. But since this device has only a
Purchases are made for both men’s and women’s suits, so that for a male suit one piece is sold, while for a female suit two pieces (top and bottom) are sold. The store has estimated that two-thirds of the swimming suits sold are for women. Let X denote the number of pieces of suits sold within
Let ????1, ????2 be two constants with 0 ≤ ????1, ????2 ≤ 1 and define the probability function f (x) by the formula f (x) = ????1f1(x) + ????2f2(x), x = 0, 1, 2,…Find the condition that must be satisfied by the ????i’s so that f is a valid probability function of a random variable, which
Let f1, f2 be the probability functions of the random variables X1, X2, which follow the Poisson distribution with parameters ????1, ????2, respectively, that isfor i = 1, f(x)=e^ x! x = 0, 1, 2,...,
The number of burglaries in a city is thought to have a Poisson distribution, and it is known that the mean of this distribution is three burglaries per week.(i) What is the probability that in a given week, there are at most two burglaries?(ii) Find the probability that in a 40-week period, there
The number of deaths at a large hospital during a month has the Poisson distribution.If the probability that at most one death occurs during a month is equal to the probability of having exactly two deaths, find the probabilities(i) to have at least one death during a month;(ii) to have at most
Razors produced by a machine are either defective, with probability 0.01, or nondefective.(i) In a batch of 200 razors, find the probability that(a) none is defective;(b) exactly two are defective using the binomial distribution.(ii) For each case in (i), find the percentage error if the Poisson
In a company with 300 employees, find the probability that exactly 4 employees have their birthday on February 14th (assuming a nonleap year of 365 days)(i) using the binomial distribution;(ii) using the Poisson approximation to the binomial.
At a zoo, the probability that a visitor requires medical attention during a day is 0.0004. Use the Poisson distribution to find the probability that, on a particular day with 3500 visitors, at least 2 will need medical attention.
The number of defects in a 100 ft long wire produced in a factory is Poisson distributed with a mean of 3.4. What is the probability that in a wire of that length,(i) there are at least three defects?(ii) there are at most three defects?(iii) there are at least five defects if, at a preliminary
Let X be a random variable such that X ∼ h(n;a, b). Show that the probability function f of X can be calculated via the following recursive scheme: f(x + 1) = (n-x)(a-x) (x+1)(bn+x+1) with the initial conditions f(x), max (0,n-b]x min{a, n}, and f(0)= f(n-b)= (h) (a+b) (nab) (a + b) n if n < b,
Jimmy likes to play a poker game with his friends once in a while. In the first hand, he receives 5 cards out of a pack of 52 cards.(i) What is the probability that he has a full house (i.e. three of a kind and two of another kind, e.g. three aces and two kings)?(ii) In the first 20 hands, he has
Bees sit on a particular flower according to a Poisson process with a rate of ???? = 6 bees per minute. What is the probability that(i) exactly four bees visit the flower in two minutes?(ii) at least one bee visits the flower in a half-minute period?
The arrival of airplanes at an airport can be modeled by a Poisson process with a rate???? = 5 arrivals per hour.(i) What is the probability that there will be at least one arrival between 3:30 p.m.and 5:00 p.m. on a particular day?(ii) Find the expected value and the variance of the number of
The number of hits to a website follows a Poisson process with a rate of 8 hits per minute. What is the probability that there will be 20 hits in a five-minute period, if we know that there have been 7 hits during the first minute of that period?
Make a graph of the hypergeometric distribution, h(n;a, b), for n = 30, a = 3k, b = 2k, where k = 50,100, 150,…, and compare this with the graph of the binomial distribution with n = 30, p = 0.6. What do you observe? Can you justify this result?
In a certain lottery, there is a total of 300 tickets and 40 of them win some amount.If we purchase 15 tickets, what is the probability that we have(i) exactly two winning tickets?(ii) at least six winning tickets?(iii) at most 10 winning tickets?
Create a large number (e.g. 10 000) of observations from the geometric distribution with parameter p = 0.10, 0.50, 0.90, 0.99. In each case, calculate the arithmetic mean of these observations. How close are these values to the theoretical mean of the distribution?
How many times do we have to throw a die so that the outcome in at least one throw is less than three and in at least one throw it is greater than two with probability more than or equal to(i) 50%; (ii) 90%; (iii) 95%; (iv) 99%.Suggestion: Work as in Example 5.6 and use Mathematica to solve the
An airline estimates that 3% of the persons who book a seat to travel do not show up. On that basis, suppose that the company makes 350 reservations for an aircraft capacity of 340 seats. What is the probability that every passenger who shows up for travel finds a seat?
As in Example 5.5, suppose Jimmy shoots against a target with the probability of hitting the target being p, and successive shootings are assumed independent. This time he is told that a prize is awarded if he chooses to shoot 2n times and hits the target at least n times. He has been offered the
Use Mathematica to draw a graph of the probability functions and the cumulative distribution functions of the seven discrete distributions given in Table 5.2, for various choices of their parameters.
The number of fraudulent credit card uses in a large store follows a Poisson process with a rate of ???? = 3 per day. The store is open 12 hours each day (except Sundays).(i) What is the expected number of frauds during a week, i.e. in six working days?(ii) If the average cost of a fraud is $240,
Nick, who is a car mechanic, receives damaged cars for repair at a rate of three cars every two hours. He has estimated that, among the cars arriving 75% are minor repairs, which he can fix by himself, while the rest should be sent to the central repair store. Find the probability that there will
.(i) Using the identityΣ∞r=0 tr∕r! = et, prove that for any real t, the following identity holds:(ii) Telephone calls arrive at a large company according to a Poisson process with a rate ???? per minute. If time is measured in minutes, and t (Hint: You may find the result in Part (i) to be
The number of trees in a forested area of S square feet follows a Poisson process with rate ????.(i) Find the probability that the distance from a given tree to the one closest to it will be at least ???? feet.(ii) Calculate the probability that the distance between a given tree to the tree which
In seismology, an earthquake is said to be “strong” if it has a magnitude of at least six measured on the Richter scale. Imagine that in an area which is frequently hit by earthquakes, the number of strong earthquakes follows a Poisson process with a rate of 2.5 per year.(i) What is the
The number of goals that Real Madrid scores in the football Champions League competition follows a Poisson process with a rate of ???? = 0.12 goals in a five-minute period. Find the probability that in their next three matches, Real will score exactly one goal in two of them and three goals in the
In the wire production unit of a factory, there is an employee who inspects the quality of the wire as it comes out of the machine that produces it. It has been estimated that the number of defects on the wire follows a Poisson process with a rate of one defect per 100 m of wire produced. One day,
Customers arrive at a bank according to a Poisson process with a rate of ???? = 2 customers for a five-minute period. Find the probability that(i) three customers will enter the bank between 2:00 p.m. and 2:15 p.m.;(ii) three customers will enter the bank between 2:00 p.m. and 2:15 p.m. and two
Irene has a Facebook page and she is very keen to have a large number of friends there. The number of friends added to her page follows a Poisson process with a rate of ???? = 3 persons per week.(i) What is the probability that on a particular week she makes less than three new friends?(ii) Find
A referendum is going to take place in an European country in order to decide whether the country will adopt the euro as its currency unit or not. Suppose that the number of eligible voters in the referendum is 15 million people, and 53% of them are in favor of adopting the euro. If 750 000
In the section that sells swimming suits at a large department store, the number of purchases in an hour follows the Poisson distribution with ???? =
6.44 Find the normal approximation to P(355x 360) for a binomial probability distribution with n 400 and p=0.9.
6.31 How Many Words? A publisher has discov- ered that the numbers of words contained in a new manuscript are normally distributed, with a mean equal to 20,000 words in excess of that specified in the author's contract and a standard deviation of 10,000 words. If the publisher wants to be almost
6.30 Loading Grain A grain loader can be set to discharge grain in amounts that are normally distributed, with mean kg and standard deviation equal to 700 kg. If a company wishes to use the loader to fill containers that hold 54,440 kg of grain and wants to overfill only one container in 100, at
6.29 Bacteria in Drinking Water Suppose the numbers of a particular type of bacteria in samples of 1 millilitre (mL) of drinking water tend to be approxi- mately normally distributed, with a mean of 85 and a standard deviation of 9. What is the probability that a given 1-mL sample will contain more
6.28 Tax Audit How does the Canada Revenue Agency decide on the percentage of income tax returns to audit for each province? Suppose they do it by ran- domly selecting 50 values from a normal distribution with a mean equal to 1.55% and a standard deviation equal to 0.45%. (Computer programs are
6.27 Economic Forecasts One method of arriving at economic forecasts is to use a consensus approach. A forecast is obtained from each of a large number of analysts, and the average of these individual forecasts is the consensus forecast. Suppose the individual 2013 January prime interest rate
6.26 Breathing Rates The number of times, X, an adult human breathes per minute when at rest depends on the age of the human and varies greatly from person to person. Suppose the probability distribution for X is approximately normal, with the mean equal to 16 and the standard deviation equal to 4.
6.25 Sunflowers An experimenter publishing in the Annals of Botany investigated whether the stem diam- eters of the dicot sunflower would change depending on whether the plant was left to sway freely in the wind or was artificially supported.' Suppose that the unsupported stem diameters at the base
6.24 A Phosphate Mine The discharge of suspended solids from a phosphate mine is normally distributed, with a mean daily discharge of 27 milligrams per litre (mg/L) and a standard devia- tion of 14 mg/L. What proportion of days will the daily discharge exceed 50 mg/L?
6.23 Elevator Capacities Suppose you must establish regulations concerning the maximum number of people who can occupy an elevator. Suppose a study of elevator occupancies indicates that, if eight people occupy the elevator, the probability distribu- tion of the total weight of the eight people has
6.22 Braking Distances For a car travelling 50 kilometres per hour (km/h), the distance required to brake to a stop is normally distributed with a mean of 15 metres (m) and a standard deviation of 2.5 m. Suppose you are travelling 50 km/h in a residential area and a car moves abruptly into your
6.21 Cerebral Blood Flow Cerebral blood flow (CBF) in the brains of healthy people is normally distributed with a mean of 74 and a standard deviation of 16.a. What proportion of healthy people will have CBF readings between 60 and 80?b. What proportion of healthy people will have CBF readings above
6.20 Christmas Trees The diameters of Douglas firs grown at a Christmas tree farm are normally distributed with a mean of 10 cm and a standard devia- tion of 3 cm.a. What proportion of the trees will have diameters between 8 and 12 cm?b. What proportion of the trees will have diameters less than 7

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