Problem 12-15 (Algorithmic)
Strassel Investors buys real estate, develops it, and resells it for a profit. A new property is available, and Bud Strassel, the president and owner of Strassel Investors, believes if he purchases and develops this property it can then be sold for $160000. The current property owner has asked for bids and stated that the property will be sold for the highest bid in excess of $100000. Two competitors will be submitting bids for the property. Strassel does not know what the competitors will bid, but he assumes for planning purposes that the amount bid by each competitor will be uniformly distributed between $100000 and $150000.
In: Statistics and Probability
Problem 16-15 (Algorithmic)
Strassel Investors buys real estate, develops it, and resells it for a profit. A new property is available, and Bud Strassel, the president and owner of Strassel Investors, believes if he purchases and develops this property it can then be sold for $160,000. The current property owner has asked for bids and stated that the property will be sold for the highest bid in excess of $100,000. Two competitors will be submitting bids for the property. Strassel does not know what the competitors will bid, but he assumes for planning purposes that the amount bid by each competitor will be uniformly distributed between $100,000 and $150,000.
In: Advanced Math
Strassel Investors buys real estate, develops it, and resells it for a profit. A new property is available, and Bud Strassel, the president and owner of Strassel Investors, believes if he purchases and develops this property it can then be sold for $160,000. The current property owner has asked for bids and stated that the property will be sold for the highest bid in excess of $100,000. Two competitors will be submitting bids for the property. Strassel does not know what the competitors will bid, but he assumes for planning purposes that the amount bid by each competitor will be uniformly distributed between $100,000 and $150,000.
a. Develop a worksheet that can be used to simulate the bids made by the two competitors. Strassel is considering a bid of $130,000 for the property. Using a simulation of 1000 trials, what is the estimate of the probability Strassel will be able to obtain the property using a bid of $130,000? Round your answer to 1 decimal place. Enter your answer as a percent.
________%
b. How much does Strassel need to bid to be assured of obtaining the property?
$_______
What is the profit associated with this bid?
$_______
c. Use the simulation model to compute the profit for each trial
of the simulation run. With maximization of profit as Strassel’s
objective, use simulation to evaluate Strassel’s bid alternatives
of $130,000, $140,000, or $150,000. What is the recommended bid,
and what is the expected profit?
A bid of $_____ results in the largest mean profit of $______.
In: Statistics and Probability
Problem 12-15 (Algorithmic)
Strassel Investors buys real estate, develops it, and resells it for a profit. A new property is available, and Bud Strassel, the president and owner of Strassel Investors, believes if he purchases and develops this property it can then be sold for $170000. The current property owner has asked for bids and stated that the property will be sold for the highest bid in excess of $100000. Two competitors will be submitting bids for the property. Strassel does not know what the competitors will bid, but he assumes for planning purposes that the amount bid by each competitor will be uniformly distributed between $100000 and $150000.
In: Math
Strassel Investors buys real estate, develops it, and resells it for a profit. A new property is available, and Bud Strassel, the president and owner of Strassel Investors, believes if he purchases and develops this property it can then be sold for $165000. The current property owner has asked for bids and stated that the property will be sold for the highest bid in excess of $100000. Two competitors will be submitting bids for the property. Strassel does not know what the competitors will bid, but he assumes for planning purposes that the amount bid by each competitor will be uniformly distributed between $100000 and $155000.
In: Operations Management
In each problem, make sure that you are clearly defining random variables, stating their distributions, and writing down the formulas that you are using. (That is, write down the pmf, write down mean and variance formulas.
4. In the video game Dota 2, the character Phantom Assassin has a 0.15 chance to land a critical hit with every attack. Assume that each attack is independent from the previous attacks. (They actually aren’t, but we’ll simplify the problem by making this assumption. See the end of assignment for an interesting discussion of the actual mechanics.) (Hint: Declare a single random variable to use for parts a through d, and a new random variable for parts e and f)
a. Suppose we start hitting a target. What is the probability that we get the first critical hit on our third attack? What is the probability that it takes more than 10 attacks to get a single critical hit?
b. How many attacks should a Phantom Assassin player expect to have to perform before getting a critical hit? What is the standard deviation?
c. It’s subjective to decide when our luck is ‘bad’, but a common definition of an event being unlikely is that it has probability less than 0.05. By this definition, how many times do we need to attack without getting a critical hit before we can say that our luck was bad?
d. Suppose we attack 20 times without getting a single critical hit. What is the probability that the next attack will be a critical hit? (Hint: Does the distribution remember that you’ve failed 20 times already? Does it care at all?)
e. Suppose Phantom Assassin attacks an enemy 30 times. What is the probability that she gets at least 3 critical hits?
f. What is the expected number of critical hits after this many attacks?
In: Statistics and Probability
Sensitivity and Specificity
We are interested in looking at the connection between a test and a disease to investigate the ability of the test to distinguish between sick and healthy.
We look at a sample of 50,000 people who have been tested for a particular disease. Of these, 100 have the disease. Of the 100 who have the disease, 95 are receiving positive test results. Of those who are healthy, there are 48902 people who get negative test results.
Set up a table showing the number of sick / healthy with positive / negative tests.
What is the probability in this sample to have the disease?
What is the sensitivity and specificity of the test and what does this mean in words?
4. What is the predictive value of positive test? What does this mean in words and what does this mean for the practical value of the test? Also, find out what the predictive value of the negative test is.Also record positive predictive value and negative predictive value using Bayes rule.
5. If we instead test vulnerable risk groups, the probability of the disease increases to 5%. The sensitivity and specificity of the test are the same as you found in the assignment. 6.3. What happens to predictive positive value if we look at 50,000 people at risk of the disease? Set up a new table and rain out PPV.
We are looking at another type of test, the probability of having the disease in vulnerable countries is 10%. The probability that the test is positive when infected with the disease is 0.999 and the probability that the test is negative when not infected is 0.99.
What is the sensitivity and specificity of the test?
7. What will be positive predictive value? Use Baye's rule.
Worldwide, the likelihood of having the disease is 1%. If the test's sensitivity and specificity are the same, what will be the positive predictive value?
Do you see a connection between the prevalence of the disease and the positive predictive value?
In: Statistics and Probability
The Monty Hall problem is a famous problem loosely based on the game show Let's Make a Deal. You are a contestant on the game show. There are 3 doors in front of you. Behind one door is a prize, and behind the other two doors are goats. Assume the door with the prize is picked uniformly at random from the three doors.
First, you pick a door. Then, Monty Hall will open one of the other two doors that you haven't picked. Monty Hall will always open a door with a goat behind it. (He knows ahead of time where the prize and goats are.) Now you have a decision to make. You can keep the door you initially picked, or you can switch to the remaining door. (One door is now open, so those are the only two doors left.)
Which is the better strategy: stay with your current door, or switch to the other door? Or are the strategies equally good?
Here's a common (incorrect, but maybe convincing) argument: There are only 2 doors remaining since Monty Hall opened one with a goat. One of them has a prize, the other has the goat. Each door has a 1/2 probability of having the prize behind it. So, in terms of your probability of winning the prize, it doesn't matter if you switch or not.
a) Let's use the sample space Ω = { 1 , 2 , 3 } to model this situation where the number indicates which door has the prize behind it. What is the probability measure for this sample space to model this situation?
b-c) If you do stay with your initial door, what is the probability that you will win the prize? If you switch to the other door, what is the probability you will win the prize? Explain a bit about your answers, to convince someone who believes the probabilities are both 1/2.
In: Statistics and Probability
55. A family has three children. Let A be the event that they have less than two girls and B be the event that they have exactly two girls.
(a) List all of the basic outcomes in A.
(b) List all of the basic outcomes in B.
(c) List all of the basic outcomes in A ∩ B
(d) List all of the basic outcomes in A U B.
(e) If male and female births are equally likely, what is the
probability of A?
56. Let B = A^c. Are A and B mutually exclusive? Are they collectively exhaustive?
67. Suppose that P(B) = 0.4, P(A|B) = 0.1 and P(A|B^c) = 0.9
(a) Calculate P(A)
(b) Calculate P(A|B)
71. Suppose a couple decides to have three children. Assume that the sex of each child is independent, and the probability of a girl is 0.48, the approximate figure in the US.
(a) How many basic outcomes are there for this experiment? Are they equally likely?
(b) What is the probability that the couple has at least one girl?
72. Let A and B be two arbitrary events. Use the addition rule and axioms of probability to establish the following results.
(a) Show that P(A U B) ≤ P(A) + P(B).
(this is called Boole’s inequality).
(b) Show that P(A n B) is greater than or equal to P(A) + P(B) - 1.
(this is called Bonferroni's Inequality)
73. Let A and B be two mutually exclusive events such that P(A) > 0 and P(B) > 0. Are A and B independent?
104. A multiple-choice quiz has 12 questions, each of which has 5 choices. To pass you need to get at least 8 of them correct. Nina forgot to study, so she simply guesses at random.
Let the random variable X denote the number of questions that Nina gets correct on the quiz. What kind of random variable is X? Specify all parameter values.
Calculate the probability that Nina passes the quiz.
In: Statistics and Probability
The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Complete parts (a) through (d) below. LOADING... Click the icon to view the table. a. Find the probability of getting exactly 1 girl in 8 births.Number of Girls x P(x) 0 0.002 1 0.028 2 0.117 3 0.217 4 0.272 5 0.217 6 0.117 7 0.028 8 0.002
In: Statistics and Probability