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Discrete Probability Distributions

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1. The probabilities that a customer selects 1, 2, 3, 4 and 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively. Fill in the probabilities.

Outcome X 1 2 3 4 5
P(X)

2. A study conducted by a TV station showed the number of televisions per household and the corresponding probabilities for each. Find the mean, variance and standard deviation

Number of televisions X 1 2 3 4
Probability P(X) 0.32 0.51 0.12 0.05

Mean:

Variance:

Standard Deviation:

3. A florist determines the probabilities for the number of flower arrangements she delivers each day. Find the mean, variance, and standard deviation for the distribution shown

Number of arrangements X 6 7 8 9 10
Probability P(X) 0.2 0.2 0.3 0.2 0.1

Mean:

Variance:

Standard Deviation:

4. A lottery offers one $1000 prize, one $500 prize, and five $100 prizes. One thousand tickets are sold at $3 each. Find the expectation if a person buys two tickets. Assume that the player’s ticket is replaced after each drzw and that the same ticket can win more than one prize. Calculate the expected value.

5. In a Gallup Survey, 90% of the people interviewed were unaware that maintaining a healthy weight could reduce the risk of stroke. If 15 people are selected at random, find the probability that at least 9 are unaware that maintaining a proper weight could reduce the risk of stroke. Show calculations.

6. In a restaurant, a study found that 42% of all patrons smoked. If the seating capacity of the restaurant is 80 people, find the mean, variance, and standard deviation of the number of smokers. About how many seats should be available for smoking customers?

Mean:

Variance:

Standard Deviation:

Answer the question.

7. In a survey, 63% of Americans said they own an answering machine. If 14 Americans are selected at random, find the probability that exactly 9 own an answering machine.. Apply the Binomial Probability Formula, page 247.

1. The probabilities that a customer selects 1, 2, 3, 4 and 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively. Fill in the probabilities.

Outcome X 1 2 3 4 5
P(X)

2. A study conducted by a TV station showed the number of televisions per household and the corresponding probabilities for each. Find the mean, variance and standard deviation

Number of televisions X 1 2 3 4
Probability P(X) 0.32 0.51 0.12 0.05

Mean:

Variance:

Standard Deviation:

3. A florist determines the probabilities for the number of flower arrangements she delivers each day. Find the mean, variance, and standard deviation for the distribution shown

Number of arrangements X 6 7 8 9 10
Probability P(X) 0.2 0.2 0.3 0.2 0.1

Mean:

Variance:

Standard Deviation:

4. A lottery offers one $1000 prize, one $500 prize, and five $100 prizes. One thousand tickets are sold at $3 each. Find the expectation if a person buys two tickets. Assume that the player’s ticket is replaced after each drzw and that the same ticket can win more than one prize. Calculate the expected value.

5. In a Gallup Survey, 90% of the people interviewed were unaware that maintaining a healthy weight could reduce the risk of stroke. If 15 people are selected at random, find the probability that at least 9 are unaware that maintaining a proper weight could reduce the risk of stroke. Show calculations.

6. In a restaurant, a study found that 42% of all patrons smoked. If the seating capacity of the restaurant is 80 people, find the mean, variance, and standard deviation of the number of smokers. About how many seats should be available for smoking customers?

Mean:

Variance:

Standard Deviation:

Answer the question.

7. In a survey, 63% of Americans said they own an answering machine. If 14 Americans are selected at random, find the probability that exactly 9 own an answering machine.. Apply the Binomial Probability Formula, page 247.

Probability

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A recent report in Business Week indicated that 20 percent of all employees steal from their company each year.

If a company employs 50 people, what is the probability that:

a. Fewer than 5 employees steal?
b. More than 5 employees steal?
c. Exactly 5 employees steal?
d. More than 5 but fewer than 15 employees steal?

Binomial Tree, CAPM, and Bond Duration

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1. The current price of IBM stock is $88. Every year the stock can either go up by 20% with probability 2/3, or go down by 10% with probability 1/3. What is the price of a European Call option on IBM with maturity in 2 years, and a strike price of $100? The risk free interest rate is 4% a year.

2. Goldseeker Inc. is a company searching for a sunken ship filled with gold at the bottom of the Atlantic. The company will have a single cashflow, two years from today. This cashflow depends only on whether the company finds the ship or not, and does not depend on any macroeconomic market factors. The probability distribution of the cashflow is given below:

probability cashflow
Ship found 0.2 $100 million
Ship not found 0.8 $0

The annual expected market return is 12%. The annual riskfree rate is 4%. The CAPM holds. What is the value of Goldseeker Inc. today?

3. A pension fund has an obligation to pay $630 million in 5.5 years. The fund invests money in a bond portfolio in order to secure this future obligation. The bond portfolio includes the following two bonds:

Bond A: Par value $1,000, coupon rate 8%, yield to maturity 6%, maturity 2 years from now (first coupon one year from now).

Bond B: Par value $1,000, coupon rate 0%, yield to maturity 6%, maturity T years from now.

The fund calculates that the investment which minimizes exposure to interest rate changes is an investment of $155 million in Bond A and $302.25 million in Bond B. What is the time to maturity, T, of Bond B?

MCQs on Transportation and probability

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1. A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl’s Sea Sprite, and the boy’s Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20. The LP model should maximize profit. There are several conditions that the store needs to worry about. One of these is space to hold the inventory. The adult bikes need two feet, but each children’s bike needs only one foot. The store has 500 feet of space. There are 1200 hours of assembly time available. The children’s bikes need 4 hours each; the Open Trail needs 5 hours and the Cityscape needs 6 hours. The store would like to place an order for at least 275 bikes.
Formulate a model for this problem.
Solve your model with any computer package available to you.
How many of each kind of bike should be ordered and what will the profit be?

Enter the profit ONLY below in the form xxxx or xxxx.0 round to nearest dollar

________________________________________________________________________

2. The following data summarizes the historical demand for a product.
Month Actual demand
March 20
April 25
May 38
June 32
July 22
August 46
Use a four period moving average and determine the forecasted demand for September

________________________________________________________________________

3. During summer weekdays, boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of 8 per hour. In a 2-hour period,
what is the probability that exactly 10 boats arrive?
Put your answer in the form 0.xxx or .xxx

________________________________________________________________________
4. The local operations manager for the IRS must decide whether to hire 1, 2, or 3 temporary workers. He estimates that net revenues will vary with how well taxpayers comply with the new tax code.
low medium high
# of workers compliance compliance compliance
1 50 50 50
2 100 60 20
3 150 70 -10
If he thinks the chances of low, medium, and high compliance are 40%, 20%, and 30%, respectively, what is his expected value of perfect information?

________________________________________________________________________
5. To find the optimal solution to a linear programming problem using the graphical method
find the feasible point that is the farthest away from the origin
find the feasible point that is at the highest location>
find the feasible point that is closest to the origin
None of the alternatives is correct
________________________________________________________________________
6. Because the dual price represents the improvement in the value of the optimal solution per unit increase in right-hand-side, a dual price cannot be negative.
True
False
________________________________________________________________________
7. The constraint 2×1 – x2 = 0 passes through the point (400,200).
True
False
________________________________________________________________________

8. A constraint with a positive slack value
will have a positive dual price
will have a negative dual price
will have a dual price of zero
has no restrictions for its dual price
________________________________________________________________________
9. A section of output from The Management Scientist is shown here.
Variable Lower Limit Current Value Upper Limit
1 60 100 120
What will happen to the solution if the objective function coefficient for variable 1 decreases by 20?
Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same
The value of the objective function will change, but the values of the decision variables and the dual prices will remain the same
The same decision variables will be positive, but their values, the objective function value, and the dual prices will change
The problem will need to be resolved to find the new optimal solution and dual price
________________________________________________________________________
10. For any constraint, either its slack/surplus value must be zero or its dual price must be zero.
True
False
________________________________________________________________________
11. There is a dual price for every decision variable in a model.
True
False
________________________________________________________________________

12. In the linear programming formulation of a transportation network (Points: 2)
there is one constraint for each node
here is one variable for each arc
the sum of variables corresponding to arcs out of an origin node is constrained by the supply at that node
All of the alternatives are correct
________________________________________________________________________
13. The main difference betwen Bayesian and Shafer-Dempster systems of probability is:
Shafer-Dempster is simply not accepted by the mathematical community.
Shafer-Dempster leaves room for future information to fill in present uncertainty.
Bayes has been discredited since it is so old.
Bayes uses past and prior probabilities for updates while Shafer-Dempster uses present and future probabilites.
None of the above are true
________________________________________________________________________
14. Use of the Poisson probability distribution assumes that arrivals are not random.
True
False
15. The feasible solution is the best solution possible for a mathematical model. (Points: 1)
True
False
_______________________________________________________________________

16. A furniture store has set aside 800 square feet to display its sofas and chairs. Each sofa utilizes 50 sq. ft. and each chair utilizes 30 sq. ft. At least five sofas and at least five chairs are to be displayed.
Suppose the profit on sofas is $200 and on chairs is $100. On a given day, the probability that a displayed sofa will be sold is .03 and that a displayed chair will be sold is .05. Mathematically model the following objective:
Maximize the total expected daily profit.
Max s + c
Max .03s + .05c
Max 6s + 5c
________________________________________________________________________

17. If P(A|B) = .2 and P(Bc) = .6, then P(B|A)
is .8
is .12
is .33
cannot be determined
________________________________________________________________________
18. The probability of a continuous variable having a specific value is 0.
True
False
________________________________________________________________________
19. The binomial distribution is appropriate to use to find the probability of the elapsed time between successes.
True
False
________________________________________________________________________
20. If x is normally distributed with mean 12 and standard deviation 2, then P(x  9) is

P(z  9/10)

P(z  -3/2)

P(z  2/3)

P(z  -3/4)

Uniform distribution and joint probabiliy

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The probabilities of the events A and B are .20 and .30, respectively. The probability that both A and B occur is .15. What is the probability of either A or B occurring?

America West Airlines reports the flight time from Los Angeles International Airport
to Las Vegas is 1 hour and 5 minutes, or 65 minutes. Suppose the actual flying time is uniformly distributed between 60 and 70 minutes.

A. Show a graph of the continuous probability distribution.
B. What is the mean flight time? What is the variance of the flight times?
C. What is the probability the flight time is less than 68 minutes?
D. What is the probability the flight takes more than 64 minutes?

Combinations and probabilities with and without replacement

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The first card selected from a standard 52-card deck was a king.

If it is returned to the deck, what is the probability that a king will be drawn on the second selection?

If the king is not replaced, what is the probability that a king will be drawn on the second selection?

What is the probability that a king will be selected on the first draw from the deck and another king on the second draw (assuming that the first king was not replaced)?

#82
A case of 24 cans contains 1 can that is contaminated. Three cans are to be chosen randomly for testing.

A. How many different combinations of 3 cans could be selected?
B. What is the probability that the contaminated can is selected for testing?