Math Stat Final
20. A person uses his car 30% of the time, walks 30% of the time and rides the bus 40% of the time as he goes to work. He is late 10% of the time when he walks; he is late 3% of the time when he drives; and he is late 7% of the time he takes the bus. a. What is the probability he took the bus if he was late? b. What is the probability he walked if he is on time?
Bayes theorem
An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects (which helps to explain why Chartair has the lowest market share). a. If an ELT is randomly selected from the general population of all ELTs, find the probability that it was made by the Altigauge Manufacturing Company. b. If a randomly selected ELT is then tested and is found to be defective, find the probability that it was made by the Altigauge Manufacturing Company.
Bayes theorem
Assume the probability of having tuberculosis (TB) is 0.0005, and a test for TB is 99% accurate. What is the probability one has TB if one tests positive for the disease?
Bayes theorem
At a certain stage in a trial, the judge feels the odds are two to one the defendant is guilty. It is determined that the defendant is left handed. An investigator convinces the judge this is six times more likely if the defendant is guilty than if he were not. What is the likelihood, given this evidence, that the defendant is guilty?
Bayes theorem
In a study of pleas and prison sentences, it is found that 45% of the subjects studied were sent to prison. Among those sent to prison, 40% chose to plead guilty. Among those not sent to prison, 55% chose to plead guilty. a. If one of the study subjects is randomly selected, find the probability of getting someone who was not sent to prison. b. If a study subject is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was not sent to prison.
Bayes theorem
On a game show, a contestant can select one of four boxes. The red box contains one $100 bill and nine $1 bills. A green box contains two $100 bills and eight $1 bills. A blue box contains three $100 bills and seven $1 bills. A yellow box contains five $100 bills and five $1 bills. The contestant selects a box at random and selects a bill from the box at random. If a $100 bill is selected, find the probability that it came from the yellow box.
Bayes theorem
Adina sets up a taste test of 3 different waters: tap, bottled in glass, and bottled in plastic. She puts these waters in identical cups and has a friend taste them one by one. The friend then tries to identify which water was in each cup. Assume that Adina's friend can't taste any difference and is randomly guessing. What is the probability that Adina's friend correctly identifies each of the 3 cups of water?
Counting problem (hint: order does matter)
Michael is taking a quiz in his music history class. The teacher writes the names of 6 songs on the board, and then plays 4 songs out of the 6, one after the other in a random sequence. Michael then needs to identify each of the songs in the sequence they were played. Suppose that Michael hasn't studied and is randomly guessing the songs. What is the probability that Michael correctly identifies all 4 songs in the correct order?
Counting problem (hint: order matters!!)
There are six men and seven women in a ballroom dancing class. If four men and four women are chosen and paired off, how many pairings are possible?
Counting problem (order does not matter but multiply genders)
A full house in poker is a hand where three cards share one rank and two cards share another rank. How many ways are there to get a full-house? What is the probability of getting a full-house?
Counting problem (order does not matter but without replacement)
20 politicians are having a tea party, 6 Democrats and 14 Republicans. To prepare, they need to choose: 3 people to set the table, 2 people to boil the water, 6 people to make the scones. Each person can only do 1 task. (Note that this doesn't add up to 20. The rest of the people don't help.) (a) In how many different ways can they choose which people perform these tasks? (b)Suppose that the Democrats all hate tea. If they only give tea to 10 of the 20 people, what is the probability that they only give tea to Republicans? (c)If they only give tea to 10 of the 20 people, what is the probability that they give tea to 9 Republicans and 1 Democrat?
Counting problem (order does not matter)
In how many ways can a committee of 5 be formed from a group of 11 people consisting of 4 teachers and 7 students if (i) the committee must include exactly 2 teachers? (ii) the committee must include at least 3 teachers? (iii) a particular teacher and a particular student cannot be both in the committee?
Counting problem (order does not matter)
Lin and Kai are friends that work together on a team of 12 total people. Their manager is going to randomly select 2 people from the team of 12 to attend a conference. What is the probability that Lin and Kai are the 2 people chosen?
Counting problem (order does not matter)
Sofie claims to be a tasting expert. To test her abilities, someone bakes Sofie a cake using 3 of the following ingredients: Coffee, Vanilla, Sea salt, Nutmeg, Cinnamon, and Ginger. Sofie's challenge is to identify which set of 3 ingredients was used in the cake. Suppose that Sofie is just randomly guessing. What is the probability that Sofie correctly identifies the set of 3 ingredients in the cake?
Counting problem (order does not matter)
In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?
Counting problem (order kind of matters)
A passcode to enter a building is a sequence of 4 single digit numbers (0-9), and repeated digits aren't allowed. Suppose someone doesn't know the passcode and randomly guesses a sequence of 4 digits. What is the probability that they guess the correct sequence?
Counting problem (order matters)
In horse racing, a trifecta is a type of bet. To win a trifecta bet, you need to specify the horses that finish in the top three spots in the exact order in which they finish. If eight horses enter the race, how many different ways can they finish in the top three spots?
Counting problem (order matters)
In how many ways can 5 boys and 3 girls be seated around a table if (i) boy B1 and girl G1 are not adjacent? (ii) no girls are adjacent?
Counting problem (order matters)
Nia is 1 of 24 students in a class. Every month, Nia's teacher randomly selects 4 students from their class to act as class president, vice president, secretary, and treasurer. No one student can hold two positions. In a given month, what is the probability that Nia is chosen as president?
Counting problem (order matters)
Ten coins are tossed simultaneously. In how many of the outcomes will the third coin turn up a head?
Counting problem (think 2^n)
There are 3 arrangements of the word DAD, namely DAD, ADD, and DDA. How many arrangements are there of the word PROBABILITY?
Counting problem (think about repeated letters)
Declan's friend Luka claims that he can read minds. To test Luka's abilities, Declan draws 5 cards without replacement from a standard deck of 52 cards. What is the probability that Luka correctly identifies all 5 cards in any order?
Counting problem (think permutations and combinations)
Suppose that Luisa randomly draws 4 cards without replacement. What is the probability that Luisa gets 2 diamonds and 2 hearts?
Counting problem (think permutations and combinations)
Suppose you want to divide a 52-card deck into four hands with 13 cards each. What is the probability that each hand has a king?
Counting problem (without replacement)
Two fair dice are rolled. a. What is the (conditional) probability that one turns up two spots, given they show different numbers? b. What is the (conditional) probability that the first turns up six, given that the sum is k, for each k from two through 12? c. What is the (conditional) probability that at least one turns up six, given that the sum is k, for each k from two through 12?
Create diagram of different outcomes
A device has probability p of operating successfully on any trial in a sequence. What probability p is necessary to ensure the probability of successes on all of the first four trials is 0.85? With that value of p, what is the probability of four or more successes in five trials?
For the probability that four independent trials are all successful to be 0.85, the probability of success for each trial must be (0.85)1/4 ≈ 0.96
In a certain population, the probability a woman lives to at least seventy years is 0.70 and is 0.55 that she will live to at least eighty years. If a woman is seventy years old, what is the conditional probability she will survive to eighty years?
Note that if B⊂A then P(AB)=P(B) use what we know about sets
Let A and B be two events. Suppose the probability that neither A or B occurs is 2/3. What is the probability that one or both occur?
Use Venn diagram
Let C and D be two events with P(C) = 0.25, P(D) = 0.45, and P(C ∩ D) = 0.1. What is P(Cc ∩ D)?
Use Venn diagram
Let: P(A)=0.55; P(AB)=0.30; P(BC)=0.20; P(Ac∪BC)=0.55; P(AcBCc)=0.15 Determine, if possible, the conditional probability P(Ac|B)=P(AcB)/P(B).
Use Venn diagram
A survey of a representative group of students yields the following information: • 52 percent are male • 85 percent live on campus • 78 percent are male or are active in intramural sports (or both) • 30 percent live on campus but are not active in sports • 32 percent are male, live on campus, and are active in sports • 8 percent are male and live off campus • 17 percent are male students inactive in sports Let A = male, B = on campus, C = active in sports. a. A student is selected at random. He is male and lives on campus. What is the (conditional) probability that he is active in sports? b. A student selected is active in sports. What is the (conditional) probability that she is a female who lives on campus?
Use Venn diagram or what we know about sets
Six students take an exam in their probability course. Their probabilities of making 90 percent or more are David (0.72 ) Mary(0.83) Joan (0.75) Hal (0.92) Sharon (0.65) Wayne (0.79) Assume these are independent events. a. What is the probability three or more make grades of at least 90 percent? b. Four or more make grades of at least 90 percent? c. Five or more make grades of at least 90 percent?
Use what we know about probabilities
Two independent random numbers between 0 and 1 are selected (say by a random number generator on a calculator). What is the probability the first is no greater than 0.33 and the other is at least 0.57?
Use what we know about probabilities
The lifetime T of a device (in hours) has the Weibull distribution with shape parameter k = 1.2 and scale parameter b = 1000. a. Find the probability that the device will last at least 1500 hours. b. Compute the failure rate function.
Weibull
A candy maker produces mints that have a label weight of 20.4 grams. Assume that the distribution of the weights of these mints is N(21.37, 0.16). Suppose that 15 mints are selected independently and weighed. Let Y equal the number of these mints that weigh less than 20.857 grams. Find P(Y ≤ 2).
X follows normal and Y follows binomial
The number of random losses in a calendar year for an individual has a Poisson distribution with mean 1. When a loss occurs, the individual loss amount is either 2 or 4, with probabilities 0.6 and 0.4, respectively. When multiple losses occur for this individual, the individual loss amounts are independent. This individual purchases an insurance policy to cover the random losses with a deductible of 1 per loss. In the next calendar year, let 𝑺 be the total payment made by the insurance company to the insured. Calculate 𝑷(𝟐≤𝑺≤𝟒).
X~Poisson(1)
Let X and Y have a trinomial distribution with n = 2, pX=¼, and pY=½. Find E(Y | x)
Y|X~Binomial(2-X, 2/3)
In Orange County, 51% of the adults are males. (It doesn't take too much advanced mathematics to deduce that the other 49% are females.) One adult is randomly selected for a survey involving credit card usage. a. Find the prior probability that the selected person is a male. b. It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of females smoke cigars (based on data from the Substance Abuse and Mental Health Services Administration). Use this additional information to find the probability that the selected subject is a male.
basic probability and Bayes theorem
By some estimates, twenty percent (20%) of Americans have no health insurance. Randomly sample n=15 Americans. Let X denote the number in the sample with no health insurance. What is the probability that more than seven have no health insurance?
binomial with n=15 and p=0.2
The probability that a planted radish seed germinates is 0.80. A gardener plants nine seeds. Let Xdenote the number of radish seeds that successfully germinate. What is the average number of seeds the gardener could expect to germinate? What is the variance and standard deviation of X?
binomial with n=9 and p=0.8
Ten customers come into a store. If the probability is 0.15 that each customer will buy a television set, what is the probability the store will sell three or more?
complement rule is useful
Three fair dice are rolled. What is the probability at least one will show a six?
complement rule is useful
The number of car accidents in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 4 per week. The number of car accidents in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8 per week. Assume that on a weekly basis, the number of accidents in one highway is independent of the number of accidents in the other highway. In one particular week, exactly 5 auto accidents took place in these two highways. What is the probability that Highway #1 had exactly 2 accidents in this particular week?
conditional probabilities and Poisson
In a survey, 85 percent of the employees say they favor a certain company policy. Previous experience indicates that 20 percent of those who do not favor the policy say that they do, out of fear of reprisal. What is the probability that an employee picked at random really does favor the company policy? It is reasonable to assume that all who favor say so.
conditional probability
The serum zinc level X in micrograms per deciliter for males between ages 15 and 17 has a distribution that is approximately normal with μ = 90 and σ = 15. Compute the conditional probability P(X > 120 | X > 105).
conditional probability and normal distribution
The "fill" problem is important in many industries, such as those making cereal, toothpaste, beer, and so on. If an industry claims that it is selling 12 ounces of its product in a container, it must have a mean greater than 12 ounces, or else the FDA will crack down, although the FDA will allow a very small percentage of the containers to have less than 12 ounces. If the content X of a container has a N(12.1, σ2) distribution, find σ so that P(X < 12) = 0.01.
find z-score and use transformation 𝑋 = μ + 𝜎×𝑍
A representative from the National Football League's Marketing Division randomly selects people on a random street in Kansas City, Missouri until he finds a person who attended the last home football game. Let p, the probability that he succeeds in finding such a person, equal 0.20. And, let X denote the number of people he selects until he finds his first success. What is the probability mass function of X?
geometric
A crate contains 50 light bulbs of which 5 are defective and 45 are not. A Quality Control Inspector randomly samples 4 bulbs without replacement. Let X = the number of defective bulbs selected. Find the probability mass function, f(X) , of the discrete random variable.
hypergeometric
A lake contains 600 fish, eighty (80) of which have been tagged by scientists. A researcher randomly catches 15 fish from the lake. Find a formula for the probability mass function of X, the number of fish in the researcher's sample which are tagged.
hypergeometric
During a particular period a university's information technology office received 20 service orders for problems with printers, of which 8 were laser printers and 12 were inkjet models. A sample of 5 of these service orders is to be selected for inclusion in a customer satisfaction survey.Suppose that the 5 are selected in a completely random fashion, so that any particular subset of size 5has the same chance of being selected as does any other subset.What then is the probability that exactly X of the selected service orders were for inkjet printers?
hypergeometric
(PMF of a sum) Suppose X and Y are independent and X ∼ Bernoulli(1/2) andY ∼ Bernoulli(1/3).Determine the pmf of X + Y.
know Bernoulli distribution
Show that if event A is independent of itself, then P(A)=0 or P(A)=1.
know Idempotent law
Suppose that X ∼ Bin(n, 0.5). Find the probability mass function of Y = 2X.
know binomial distribution
Suppose that X is a random variable that has the probability function P(X=k)={0.3if k=8, 0.2if k=10, 0.5if k=6} What is the moment generating function for X?
know how to find a moment generating function [E(e^tx)]
Let X and Y have the joint pmf defined by f (0, 0) = f (1, 2) = 0.2, f (0, 1) = f (1, 1) = 0.3. (a) Depict the points and corresponding probabilities on a graph. (b) Give the marginal pmfs in the "margins." (c) Compute 𝜇,𝜇, 𝜎ଶ, 𝜎ଶ, Cov(X,Y), and ρ.
know how to find p and covariance
What is the Var(X) when X ~ Hypergeometric (n, N1, N2)?
look at book or summation sheet
Suppose the average amount of cars passing on a street per minute is Poisson (8.6). It takes a lady 5 seconds to cross a street and she waits for two cars to pass. What is the probability that she is safe to cross the street?
look at the exponential distribution
Suppose that the moment generating function of the random variable X is M (t)=(1+3e^t/4)^10. What is the mean and variance of X?
look for corresponding moment generating functions in book
The torque required to remove bolts in a steel plate is rated as very high, high, average, and low, and these occur about 30%, 40%, 20%, and 10% of the time, respectively. Suppose n = 25 bolts are rated; what is the probability of rating 7 very high, 8 high, 6 average, and 4 low? Assume independence of the 25 trials.
multinomial?
A frequent force of mortality used in actuarial science is λ(w) = aebw + c. Find the cdf and pdf associated with this Makeham's law.
see pages 119-120
Suppose that the cdf of X is given by: {F (a)={0for a<0, 1/5for 0≤a<2, 2/5for 2≤a<4, 1for a≥4} Determine the pmf of X.
see solutions week 4 number 5
An actuary performs a claim frequency study on a group of auto insurance policies. She finds that the probability function of the number of claims per week arising from this set of policies is 𝑷[𝑵=𝒏] where 𝒏= 𝟏, 𝟐, 𝟑, ..... What is the weekly average number of claims arising from this group of insurance policies?
the random variable N is a zero-truncated Poisson
In a group there are 𝐌𝐌 men and 𝐖𝐖 women; 𝒎𝒎 of the men and 𝒘𝒘 of the women are college graduates. An individual is picked at random. Let A be the event the individual is a woman and B be the event he or she is a college graduate. Under what condition is the pair {A,B} independent?
to check for independence: P(A|B) = P(A)
Suppose that the Moment generating function for X is M=e^t/(3−2e^t). Then determine μ and σ2 for X.
use E(x)
Show that if X has probability distribution Poisson(μ), then μ = E[X] = Var[X].
use MGFs (hint: M'(t) and M''(t)
In an industrial engineering article, the authors suggest using a Weibull distribution to model the duration of a bake step in the manufacture of a semiconductor. Let T represent the duration in hours of the bake step for a randomly chosen lot. Suppose T ∼ Weibull(δ = 10, β = 0.3). a. What is the probability that the bake step takes longer than four hours? b. What is the probability that the bake step takes between two and seven hours?
use Weibull function in excel
Suppose X is a random variable with E(X) = 5 and Var(X) = 2. What is E(X2)?
use algrbra
Let 𝑓(𝑥,𝑦) = (3/16)𝑥𝑦ଶ, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, be the joint pdf of X and Y. (a) Find 𝑓(𝑥) and 𝑓(𝑦), the marginal probability density functions. (b) Are the two random variables independent? Why or why not? (c) Find P(X ≤ Y).
use integrals to find pdfs, independence: f(x,y)=fx(x)fy(y)
An insurance portfolio consists of four policyholders. Assume that the number of claims for one policyholder is independent of the number of claims for any one of the other policyholders in the portfolio. What is the probability that the total number of claims in this portfolio in the upcoming calendar year is 4?
use pmf given in week 6 question 4, either negative binomial or sum of 4 geometric
Suppose that Y is a random variable with moment generating function H(t). Suppose further that X is a random variable with moment generating function M(t) given by M (t)=1/3(2e^3t+1)H (t). Given that the mean of Y is 10 and the variance of Y is 12, then determine the mean and variance of X
use product derivative rule
Directly from the definitions of expected value and variance, compute E(X) and Var(X) when X has probability mass function given by the following table:
use table from week 4 problem 1
A fair six-sided die is rolled until each face is observed at least once. On the average, how many rolls of the die are needed?
use what we know about probabilities (see week 6 question 1)
The Gallup organization randomly selects an adult American for a survey about credit card usage. Use subjective probabilities to estimate the following. a. What is the probability that the selected subject is a male? b. After selecting a subject, it is later learned that this person was smoking a cigar during the interview. What is the probability that the selected subject is a male? c. Which of the preceding two results is a prior probability? Which is a posterior probability?
use what we know about probability and Bayes
