I. Introduction
Arrowmark Vending has the contract to supply pizza at football games for a university. The operations manager, Tom Kealey, faces the challenge of determining how many pizzas to make available at the games. We have been provided with demand distributions for pizza based on past experience and know that Tom will only supply plain cheese and pepperoni and cheese combo pizzas. We also know that there is a fixed cost of $1,000 allocated equally between the two types of pizzas, and that the costs to make plain cheese pizza and pepperoni and cheese pizza are $4.50 and $5.00 respectively. Both pizzas sell for $9.00 and unsold pizzas have no value. The purpose of this report is to provide Tom with some information regarding how many of each type of pizza he should produce if he wants to achieve the highest expected profit from pizza sales at the game.

II. Analysis
In order to determine at which production level Tom will achieve the highest expected profit, it is first necessary to determine the potential profit or loss associated with producing at each demand level. To do this, a discrete probability distribution is composed for each potential level of production. For example, if 200 plain cheese pizzas are produced and 200 are demanded, the potential profit is $400. This profit consists of $1800 in sales revenue minus $1400 in costs ($900+$500 fixed). This profit will result regardless of whether more than 200 are demanded. Accordingly, if 400 cheese pizzas are produced and only 200 demanded, there is a potential loss of $500.

Using these distributions, we are then able calculate the distribution’s mean, which is the expected value of the profits at each level of production. The expected profits in this case are the weighted average of the potential profit values, in which the weights are the probabilities. The expected profits associated with each type of pizza are provided in the tables below:

...APStatistics Cole Rogers
Unit 7 Exam RandomVariables: Free Response
Directions: Complete the assignment on this paper. If you need additional paper make sure that you clearly label each page with your name. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit.
1. A Roulette wheel has 38 slots numbered 0 to 36 and 00. The wheel is spun and a ball is thrown into the wheel and comes to rest in one of the slots. There are numerous of ways to bet, individual numbers, groups of numbers (1-12, 13-24, etc), by color (half of the numbers are black and the half are red), and in various other combinations. This problem is going to focus on betting $1.00 on the number group 1-12. If the ball lands in any of the values 1-12 the bet is won and the return is $3.00. If the ball lands on any of the other values the bet is lost.
a.) Compute the expected value of this game. (4 points)
X
2
-1
P(x)
12/38
26/38
X*P(x)
.631
-.684
-.053
The expected value is -.053.
b.) Interpret this expected value. (4 points)
This means that the casino wins 5.3 cents per game or the players lose 5.3 cents per game.
c.) What is the average return to the casino from 1,000,000 such bets? (4 points)
1000000*.05=50000
The casino can expect to win 500000 of the 1 million bets because 1 million times .05 is 50,000....

...SIDS31081 - Statistics Refresher
2006 – 2007
Exercises
(Probability and RandomVariables)
Exercise 1
Suppose that we have a sample space with five equally likely experimental outcomes :
E1,E2,E3,E4,E5.
Let
A = {E1,E2}
B = {E3,E4}
C = {E2,E3,E5}
a. Find P(A), P(B), P(C).
b. Find P(A U B) . Are A and B mutually exclusive?
c. Find Ac, Bc, P(Ac), P(Bc).
d. Find A U Bc and P(A U Bc)
e. Find P(B U C)
Exercise 2
A committee with two members is to be selected from a collection of 30 people, of whom 10
are males and 20 are females.
a. Find the probability that both members are male
b. Find the probability that both members are female
c. Find the probability that one member is male and one is female.
Exercise 3
A warehouse contains 100 tires, of which 5 are defective.
Four tires are chosen at random for a new car.
Find the probability that all four are good.
Exercise 4
In a particular city,
40% of the people subscribe to magazine A, 30% of the people subscribe to magazine B and
50% to magazine C.
However, 10% subscribe to both A and B, 25% subscribe to both A and C, 15% subscribe to
both B and C. Finally, 5% subscribe to all three magazines.
A person is chosen at random.
a. What is the probability that the chosen person subscribes to at least one magazine?
b. What is the probability that the chosen person subscribes to at least two magazines?
c. Find the conditional probability that...

...The number of possible combinations of 3 horses winning, in any order, is
So the probability is
4. In how many distinguishable ways can the letters in the word statistics be written?
5. The table shows the results of a survey that asked 2850 people whether they are involved in any type of charity work.
| Frequently | Occasionally | Not at all | Total |
Male | 221 | 456 | 795 | 1472 |
Female | 207 | 430 | 741 | 1378 |
Total | 428 | 886 | 1536 | 2850 |
A person is selected at random from the sample.
a) What is the probability the person is female or occasionally involved in charity work?
b) Are the events “being female and occasionally involved in charity work” and “being frequently involved in charity work” mutually exclusive?
yes
6. A company gave psychological tests to perspective employees. The randomvariable x represents the possible test scores.
a) Use the histogram to find the probability that a person selected at random from the survey’s sample had a test score of more than two.
b) Find the probability that the person had a test score of at most 2.
7. The following table is a frequency distribution for the number of dogs per household in a small town.
Dogs | 0 | 1 | 2 |
Households | 931 | 297 | 180 |
a) Construct the probability distribution. (round to the thousandths place)
x | 0 | 1 | 2 |
P(x) | 931/1408=0.661 | 297/1408=0.211 |...

...
Event A is rolling a die and getting a 6. Suggest another event (Event B) that would be independent from Event A.
A company runs 3 servers, each providing services to 40 computers. For each server, two of its client computers are infected. What is the probability that 3 randomly chosen client computers serviced by different servers (one per server) will all be infected?
The probability that Alice’s RSA signature on a document is forged is () What is the probability that out of 4 messages sent by Alice to Bob at least one is not forged?
Event A is selecting a “red” card from a standard deck at random. Suggest another event (Event B) that is compatible with Event A.
What is the probability of getting 6 tails in 10 trials of tossing a coin? Solve this problem by using :The approximation mentioned in Theorem 6
The Binomial Distribution
Then compare answers for a) and b) after you have solved the problem.
When transmitting messages from a point A to a point B, out of every 40 messages 6 need to be corrected by applying error correcting codes. What is the probability that in a batch of 200 messages sent from A to B, there will be between 38 and 42 messages that will have to be corrected? Please choose the appropriate method to approximate this quantity.
The probability of an event occurring in each of a series of independent trials is . Find the distribution function of the number of occurrences of in 9 trials. That is, provide a...

...THE MOMENTS OF A RANDOMVARIABLE
Definition: Let X be a rv with the range space Rx and let c be any known constant. Then the kth moment of X about the constant c is defined as
Mk (X) = E[ (X c)k ]. (12)
In the field of statistics only 2 values of c are of interest: c = 0 and c = . Moments about c = 0 are called origin moments and are denoted by k, i.e., k = E(Xk ), where c = 0 has been inserted into equation (12). Moments about the population mean, , are called central moments and are denoted by k, i.e, k = E[ (X )k ], where c = has been inserted into (12).
STATISTICAL INTERPRETATION OF MOMENTS
By definition of the kth origin moment, we have:
k =
(1) Whether X is discrete or continuous, 1 = E(X) = , i.e., the 1st origin moment is simply the population mean (i.e., 1 measures central tendency).
(2) Since the population variance, 2, is the weighted average of
deviations from the mean squared over all elements of Rx, then 2 =
E[(X )2] = 2. Therefore, the 2nd central moment, 2 = 2, is a measure of dispersion (or variation, or spread) of the population. Further, the 2nd central moment can be expressed in terms of origin moments using the binomial expansion of (X )2, as shown below.
2 = E[ (X )2] = E[(X2 2 X + 2 )] = E(X2) 2 E(X) + 2
= E(X2) 2 = ()2 = 2 . (13)
Example 24 (continued). For...

...Individual Work unit-8
Section 6.1
8. a) The time it takes for a light bulb to burn out is a continuous randomvariable because the time is being measured. All possible results for the variable time (t) would be greater than > 0.
b) The weight of a T-bone steak is a continuous randomvariable because the weight of the steak is measured. All the possible results for the weight of the T-bone steak would be positive numbers making the variable weight (w) > greater than 0.
c) The number of free throw attempts before the first shot is made is a discrete randomvariable because every shot is attempt can be counted. Let (x) represent shot attempts, all the possible results of the value x would be x = 0, 1, 2, 3, 4
d) In a random sample of 20 people the number with type A blood is a discrete randomvariable because the people with type A blood are being counted. Let (x) represent people with Type A blood, all possible results of the value x would be x = 0, 1, 2
12. les; because Px=1 and 0≤Px ≤1 for all x.
16. No, because P x=1.25 ≠1.
20. a) This is a discrete probability distribution because the sum of the probabilities is 1 and the probabilities are between 0 and 1.
c) mx = x ∙Px=0 0.073+10.117+20.258+30.322+40.230=2.519=2.5. Or average the number of activities...

...Friday. Find the attendant’s expected earnings for this particular period.
4.7 By investing in a particular stock, a person can make a proﬁt in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain?
4.10 Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point
scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table
gives the joint distribution for X and Y .
4.12 If a dealer’s proﬁt, in units of $5000, on a new automobile can be looked upon as a randomvariable
X having the density function
fx= 21-x,0<x<10,elsewhere
ﬁnd the average proﬁt per automobile.
4.14 Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function.
fx= 2(x+2)50<x<10,elsewhere
4.28 Consider the information in Exercise 3.28 on page 93. The problem deals with the weight in ounces
of the product in a cereal box, with
fx= 25,23.75 ≤x ≤26.250,elsewhere.
4.33 Use Deﬁnition 4.3 on page 120 to ﬁnd the variance of the randomvariable X of Exercise 4.7 on page
117.
4.7 By investing in a particular stock, a person can make a proﬁt in one year of $4000 with probability 0.3 or take a loss of $1000 with...

...Practice problem – V
Week 6
Four of the five sugars listed below are related as members of the same subgroup. Select the exception by indicating characteristic(s) of each option in the space provided thereby showing how the exception was determined. (3 marks)
a. glucose
b. fructose
c. cellulose
d. ribose
e. deoxyribose
Four of the five sugars listed below are related as members of the same subgroup. Select the exception by indicating characteristic(s) of each option in the space provided thereby showing how the exception was determined. (3 marks)
a. lactose
b. sucrose
c. maltose
d. table sugar
e. fructose
Answer the following questions using the images supplied.
a) What type of polymer is being illustrated in the diagram and what is the evidence that lead to your decision? (2 marks)
b) Label the two types of bonds illustrated in the diagram. (2 marks)
c) What two types of structures of this polymer are being shown in the diagram? To support your answer, label the diagram in which the types of structure are illustrated. (2 marks)
d) State the specific name of the structure shown. (1 mark)
Complete the following table.
| |aldohexose...