4. A group of individuals containing b boys and g girls is lined up in random order?that is, each of the (b + g)! permutations is assumed to be equally likely. What is the probability that the person in the ith position, 1 ≤ i ≤ b+g, is a girl?
An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be (a) of the same color; (b) of different colors? Repeat under the assumption that whenever a ball is selected, its color is noted and it is then replaced in the urn before the next sel
A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability 0.7. whereas when the defendant is, in fact, innocent, this probability drops to 0.2. If 70 percent of defendants are guilty
40. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already know
Independent trials that result in a success with probability p are successively performed until a total of r successes is obtained. Show that the probability that exactly n trials are required. See the attached file.
Prove or give a counterexample. If E1 and E2 are independent, then they are conditionally independent given F. Please see the attached file for the fully formatted problem.
An absent-minded nurse is to give Mr. Brown a pill each day. The probability that the nurse forgets to administer the pill is 2/3. If he receives the pill, the probability that Mr. Brown will die is 1/3. If he does not get his pill, the probability that he will die is 3/4. Mr. Brown dies. What is the probability that the nurse f
An experimental test to detect a particular type of cancer indicates the presence of cancer in 90% of individuals known to have this type of cancer, and in 15% of individuals known to be cancer-free (false positive). One hundred individuals volunteer to take the test. Of the 100, 60 are known to have the cancer, and 40 are known
12. Let E, F, and G be three events. Find expressions for the events so that of E, F, and G: (a) only E occurs; (b) both E and G but not F occur; (c) at least one of the events occurs; (d) at least two of the events occur; (e) all three occur; (f) none of the events occurs.
19. A retail establishment accepts either the American Express or VISA credit card. A total of 24 percent of its customers carry an American Express card, 61 percent carry a VISA card, and 11 percent carry both. What percentage of its customers carry a credit card that the establishment will accept?
This problems is from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much. 18. 45% of the students at U of M do neither have a tattoo nor a body piercing (other than an ear piercing). 40% of the students at U of M ha
The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much.
1. In how many ways can 10 people sit in a row if a. there are no restrictions b. persons A and B must sit next to each other c. there are 4 men and 6 women and the 4 men must sit next to each other d. there are 5 sets of twins and each set of twins must sit together e. there are 5 men and 5 women and no 2 men or 2 women ca
Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much. 2. Twenty workers are to be assigned to 20 different jobs, one to each job. How many different assignments are possible? 3. A student is to answer 7 out of 10 questions in an examination. How many choices has sh
Suppose E,F are subsets of the sample space of an experiment with random outcomes of an experiment. We often call E,F events. Define what it means for E,F to be independent. (Question also included in attachment)
Your opponent specifies 3 successives results of tosses of a coin, e.g. HHT. You then specify another such result, e.g. THT. The winner is the person whose sequence appears first when a fair coin is tossed successively and independently. Find the strategy which will allow you, the second player, to win at least 2/3 of the time.
Assuming boy and girl children are equally likely and births are independent, if parents have two children, what is the probability that at least one is a boy? On the condition that at least one is a boy, what is the probability that one of the children is a girl?
Suppose a pair of dice are flipped. Find the probability for each of the following events: 1. The sum of the dots is even. 2. The sum of the dots is at least 5.
Please use words to describe the solution process. (I've attached a possibly useful review of probability). Suppose that P is finitely additive on an alegbra {field} A. Show that P is countably additive on A <---> whenever {An} is a sequence in ... *see attachment*
Problem 8. Suppose that an insurance company classifies people into low, average and high risk persons. Their records indicate that the probabilities of being involved in an accident over a 1-year period are 0.05, 0.15 and 0.30 for low, average and high risk persons, respectively. Assume that 30% of the population is low-risk, 5
Prove Boole's inequalities.
A random number of dice is rolled. Find the probability that... (SEE ATTACHED)
#18. Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the first card is an ace, or the second a deuce, or the third a three, or ..., or the thirteenth a king, or fourteenth an ace, and so on, we say that a match occurs. Note that we do not require that the (13n + 1)th card be any particular ac
The number of winter storms in a good year is Poisson random variable with mean 3, whereas the number in a bad year is a Poisson random variable with mean 5. If next year will be a good year with probability 0.4 or a bad year with probability 0.6, find the expected value and variance of the number of storms that will occur.
#53. A prisoner is trapped in a cell containing 3 doors. The first door leads to a tunnel that returns him to his cell after 2 days travel. The second leads to a tunnel that returns him to his cell after 4 days travel. The third door leads to freedom after 1 day of travel. If it is assumed that the prisoner will always select do
1) Suppose that 3 balls are chosen without replacement from an urn consisting of 5white and 8 red balls. Let Xi equal 1 if the ith ball selected is white, and let it equal 0 otherwise. Give the joint probability mass function of (a) X1, X2; (b) X1, X2, X3. 2) A bin o
#39. If X is an exponential random variable with parameter λ = 1, compute the probability density function of the random variable Y defined by Y = log X. #40. If X is uniformly distributed over (0,1), find the density function of Y = e^x.
34. Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked as an exponential random variable with parameter 1/20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 2
# 48. It is known that diskettes produced by a certain company will be defective with probability 0.01, independently of each other. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of 10 diskettes in the package will be defective. If someone buys 3 packages, what is the pro
You have $1000 and a certain commodity presently sells for $2 per ounce. Suppose that after one week the commodity will sell for either $1 or $4 an ounce, with these two probabilities being equally likely. (a) If your objective is to maximize the expected amount of money that you possess at the end of the week, what strategy