a.  Suppose that 33% of American CEO's are women. Furthermore, suppose that 17% of American CEO's are women under the age of 40. Given that a randomly selected American CEO is a woman, what is the probability that she is under the age of 40?

Round your answer to three decimal places.

Probability =

b. The probability that the head of a U.S. household has a life insurance policy is 0.640. Moreover, the probability that the head of a U.S. household has a life insurance policy and is over the age of 50 is 0.400. Given that a randomly selected head of a U.S. household has a life insurance policy, what is the probability that he/she is over the age of 50?

Round your answer to three decimal places.

Probability =

c. Suppose that 33% of customers purchase peanut butter during a particular trip to the grocery store. Furthermore, 18% of grocery store customers purchase both peanut butter and jelly. Given that a random grocery store customer purchases peanut butter, what is the probability that he/she also purchases jelly during this trip?

Round your answer to three decimal places.

Probability =

d. In a particular convenience store, the probability that a customer will purchase beer is 0.380. Moreover, given that the customer has purchased beer, the probability that he/she will purchase pretzels is 0.280. What is the probability that a random customer in this convenience store will purchase beer and pretzels together?

Round your answer to three decimal places.

Probability =

23. a. Suppose that 42% of American CEO's are women. Furthermore, suppose that 24% of American CEO's are women under the age of 40. Given that a randomly selected American CEO is a woman, what is the probability that she is under the age of 40?

Round your answer to three decimal places. Probability = ????

b. The probability that the head of a U.S. household has a life insurance policy is 0.510. Moreover, the probability that the head of a U.S. household has a life insurance policy and is over the age of 50 is 0.420. Given that a randomly selected head of a U.S. household has a life insurance policy, what is the probability that he/she is over the age of 50?

Round your answer to three decimal places. Probability = ????

c. Suppose that 21% of customers purchase peanut butter during a particular trip to the grocery store. Furthermore, 19% of grocery store customers purchase both peanut butter and jelly. Given that a random grocery store customer purchases peanut butter, what is the probability that he/she also purchases jelly during this trip?

Round your answer to three decimal places. Probability = ????

d. In a particular convenience store, the probability that a customer will purchase beer is 0.360. Moreover, given that the customer has purchased beer, the probability that he/she will purchase pretzels is 0.200. What is the probability that a random customer in this convenience store will purchase beer and pretzels together?

Round your answer to three decimal places. Probability =????

An person has applied for positions at Company A, Company B and Company C. The probability of obtaining an offer from Company A is 0.3, from Company B is 0.6 and from Company C is 0.8. Assume that the three job offers are independent.

A)What is the probability that the person will receive a job offer from ALL three companies?

B)What is the probability that the person will receive a job offer from Company A only ?

C)What is the probability that the person will receive job offer from Company B only ?

D)What is the probability that the person will receive job offer from Company C only ?

E)What is the probability that the person will receive a job offer from exact one company (either A, B or C)?

F)What is the probability that the person will receive a job offers from Company A and B but not C?

G)What is the probability that the person will receive a job offers from Company A and C but not B?

H)What is the probability that the person will receive a job offers from Company B and C but not A?

I)What is the probability that the person will receive a job offers from exactly two companies?

K)What is the probability that the person will not receive any job offers?

L)What is the probability that the person will receive at least one job offer (i.e. either 1 or more)?

(PLZ ANSWER ASAP)

The manager of the hospital blood bank in Lancaster receives daily requests for a rare blood type from two hospitals, one in Lancaster and one in Morecambe. Requests for this blood are known to occur at random, at different rates from the two hospitals. The request rates per day from the two hospitals are given in the table below.

1. What is the probability distribution of the number of requests for the blood in a 1-week period (i.e. 7 days) from Lancaster? [There is no need to calculate any probabilities.] Justify your answer.

2. What is the probability of exactly 4 requests from Lancaster in a 1-week period? [Do not use tables and show your working].

3. What is the probability of more than 4 requests from Lancaster in a 1-week period? [You may use tables, but show your working.]

4. What is the probability of more than 4 requests from Lancaster occurring in exactly three out of eight 1-week periods. [Do not use tables and show your working]. Justify your answer.

5. Given that requests from Lancaster and Morecambe are independent, what is the probability of a total of exactly six requests occurring in a 1-week period across both Lancaster and Morecambe? Justify your answer carefully.

Part (b)        

When banks are lending money to companies there is always a chance that the company will become bankrupt and the bank will lose the money that it has lent. To better understand the risks they are facing banks often study historical data. In one such study the company performance was categorised each year as poor (P), good (G) or excellent (E), and the following probabilities of becoming bankrupt in the following year were obtained:

P(company bankrupts next year| performance is poor) = 0.25

P(company bankrupts next year| performance is good) = 0.10

P(company bankrupts next year| performance is excellent) = 0.05

The current performance of companies is distributed 10% ‘poor’, 60% ‘good’ and 30% ‘excellent’.

1. Assuming that the historic pattern of performance continues, what is the probability that a company that bankrupts next year is one that had an ‘excellent’ performance this year, i.e.

P(current performance is excellent | company bankrupts next year)?

2. Describe briefly two ways in which a credit crisis might impact on the calculation you performed to answer part (i).

In general, high school and college students are the most pathologically sleep-deprived segment of the population. Their alertness during the day is on par with that of untreated narcoleptics and those with untreated sleep apnea. Not surprisingly, teens are also 71 percent more likely to drive drowsy and/or fall asleep at the wheel compared to other age groups. (Males under the age of twenty-six are particularly at risk.) The accompanying data set represents the number of hours 25 college students at a small college in the northeastern United States slept and is from a random sample. Enter this data into C1 of Minitab Express. 6 7 6 7 6 7 7 7 8 6 6 6 8 8 8 5 4 6 7 8 5 8 7 6 7 For the analyses that follow, we shall use 90%, 95%, and 99% as the confidence levels for the confidence interval. 5% as the level of significance (?) for the hypothesis test. 7 hours sleep as the null hypothesis (according to The Sleep Foundation). Use Minitab Express to: (i) create a boxplot – GRAPHS, Boxplot - and (ii) normal probability plot – GRAPHS, Probability Plot, and (iii) calculate descriptive statistics - STATISTICS, Describe, Descriptive Statistics. Under the “Descriptive Statistics” dialog window, click on the Statistics tab and check only Mean, SE of mean, Standard deviation, and N. Include these with the submission of your project

Use Minitab Express to: (i) create a boxplot – GRAPHS, Boxplot - and (ii) normal probability plot – GRAPHS, Probability Plot, and (iii) calculate descriptive statistics - STATISTICS, Describe, Descriptive Statistics. Under the “Descriptive Statistics” dialog window, click on the Statistics tab and check only Mean, SE of mean, Standard deviation, and N. Include these with the submission of your project.

Katharine Rally is the vice president of operations for the XYZ Company. She oversees operations at a plant that manufactures components for hydraulic systems. Katharine is concerned about the plant’s present production capability. She has reduced the decision situation to three alternatives. The first alternative, which is fully automation, would result in significant changes in present operations. The second alternative, which is semi-automation, involves fewer changes in present operations. The third alternative is to make no changes (do nothing).

As a manager of the plant management team, you have been assigned the task of analyzing the alternatives and recommending a course of action. Based on the past data, Katharine is further convinced that the capital investment, annual revenue, useful lives, and salvage values can be considered random variables with the following specified probability distributions. She also asks you to develop a simulation of 50 sample points of AW values at a MARR 0f 20%/year. Interpret your results and indicate which alternative should be selected.

Use the Random Number Generation (RNG) Data Analysis Tool package of Microsoft Excel. The online help function explains how to initiate and use the RNG to generate random numbers from a variety of probability distributions: normal, uniform (continuous variable), binomial, Poisson, and discrete.

Statically show that one of the alternatives is more appropriate than the other one using hypothesis testing?

Alternative

--------------------------------------------------------------------------------------------

Parameter                  A                                                         B                    

--------------------------------------------------------------------------------------------

Capital                         Normal                                    Normal

Investment                  Mean: $300,000                      Mean: $85,000           

                                    Std. dev.: $50,000                   Std. dev.: $500

Annual                         Normal                                    Normal                       

Revenue                      Mean: $150,000                      Mean: $85,000           

                                                Std. dev.: $10,000                   Std. dev.: $500

Useful live                   Discrete uniform                     Discrete uniform        

3 to 8 years with                     3 to 7 years with

equal probability                     equal probability        

Salvage Value             Uniform                                  Uniform         

                                                30,000 to $60,000                   $10,000 to $20000                 

Use for Questions 1-7:

Hector will roll two fair, six-sided dice at the same time. Let A = the event that at least one die lands with the number 3 facing up. Let B = the event that the sum of the two dice is less than 5.

1. What is the correct set notation for the event that “at least one die lands with 3 facing up and the sum of the two dice is less than 5”?

2. Calculate the probability that at least one die lands with 3 facing up and the sum of the two dice is less than 5.

3. What is the correct set notation for the event that “at least one die lands with 3 facing up if the sum of the two dice is less than 5”?

4. Calculate the probability that at least one die lands with 3 facing up if the sum of the two dice is less than 5.

5. What is the correct set notation for the event that “the sum of the two dice is not less than 5 if at least one die lands with 3 facing up”?

6. Calculate the probability that the sum of the two dice is not less than 5 if at least one die lands with 3 facing up. 7. Are A and B independent? Explain your reasoning.

Use for question 10: A particular type of scan is used to try to determine whether brain tumors are cancerous or not. Each time a tumor is scanned, the result is reported as either “positive”, “negative” or “inconclusive”. Among tumors that are cancerous, 68% of scans are “positive”, 28% of scans are “inconclusive”, and 4% of scans are “negative”. Among tumors that are NOT cancerous, 60% of scans are “negative”, 37% of scans are “inconclusive” and 3% of scans are “positive”. Historically, among all brain tumors, 67% are not cancerous.

10. If a tumor is scanned and the result is labeled as “inconclusive” what is the probability that the tumor is not cancerous?

Good performance (obtaining a grade of A+) in this probability class depends on your attendance (A) and completion of assignments (C). The probability that you will receive a grade of A+ are 95%, 75%, 50%, and 0%, if you attend the class and complete the assignments, if you attend but do not complete assignments, if you do not attend but complete assignments, and if you neither attend nor complete assignments, respectively. Further assume that if you attend the class, there is a 90% probability that you will complete the assignments. The probability that you will attend the class is 0.95 and the probability that you will complete the assignments is 0.90.

(a) What is the probability that you will receive an A+ in this class?

(b) If a student receives an A+, what is the probability that you attend the class and completed the assignments?

#include <iostream>
#include <queue> //This contains the STL's efficient implementation of a priority queue using a heap
using namespace std;


/*
In this lab you will implement a data structure that supports
2 primary operations:
        - insert a new item
        - remove and return the smallest item
A data structure that supports these two operations is
called a "priority queue".  There are many ways to
implement a priority queue, with differernt efficiencies
for the two primary operations.  For this lab,
please do not attempt (at least at first) to implement
something fancy (like a heap).  Just use your mind to
do something simple.  Analyze the efficiency of both
insertion and removal for your implementation.
Afterwards, feel free to investigagte "min heaps"
to see a very clever and fast priority queue
implementation.  Feel free to try to implement it.

After implementuing your priority queue, use it to
implement a sorting algorithm (priorityQueueSort).
Analyze the run time of your sorting algorithm.

Next, plug-in the stand template libraries priority queue
(which is implemented with a heap) and compare the speed.

(don't forget to run in release mode)

To get lab credit, finish all of the coding below,
as well as be able to state the run times of
your priority queue operations and sorting algorithm
to the lab TAs.
*/


//What is the big-oh run time of this sorting routine with respect
//to the number of items n?
//How does this compare to bubble sort or selection sort?
void priorityQueueSort(int * numbers, int size)
{
        priorityQueue PQ;

        //Step 1:  insert each item from 'numbers' into PQ.

        //Step 2:  Extract from PQ until PQ is empty, each time placing the extracted item into the numbers array, one after another.

}

//For this part you will need to use the built in priority queue in the STL libary.
//The STL priority_queue is implemented using a "heap".  Feel free to read about this on your own
//to understand why it is so fast.
//The STL priority_queue has the following methods:
// push(x), which adds x to the priority queue (this is like your "insert" method)
// pop(), which removes the highest value item from the priority queue
// top(), which returns the highest value item in the priority queue (but does not remove it).
// Run time:  Inserting 1 item into a min heap takes O(log n) time.  Extracting the biggest takes O(log n).
// Therefore, the run time of this sorting algorithm is O(n log n), and that is why it sorts a billion items so fast.
// Which "fast" sort is better?  Blaze sort (aka quick sort), or heap sort?
void heapSort(int * numbers, int size)
{
        priority_queue<int> PQ;

        //Step 1:  insert each item from 'numbers' into PQ.

        //Step 2:  Extract from PQ until PQ is empty, each time placing the extracted item into the numbers array, one after another.

}


int main()
{
        //Part 1:  Implement a priority queue data structure
        priorityQueue pq;

        pq.insert(59);
        pq.insert(12);
        pq.insert(548);
        pq.insert(45);
        pq.insert(18);
        pq.insert(345);

        cout << "Extracting min: " << pq.extractMin() << endl; //12
        cout << "Extracting min: " << pq.extractMin() << endl; //18
        cout << "Extracting min: " << pq.extractMin() << endl; //45
        cout << "Extracting min: " << pq.extractMin() << endl; //59

        pq.insert(2);
        pq.insert(400);
        pq.insert(600);
        pq.insert(20);


        //2 20 345 400 548 600
        while (!pq.empty())
        {
                cout << "Extracting min: " << pq.extractMin() << endl;
        }


        //Part 2:  create a sorting function that uses your priority queue data structure to sort
        int numbers[] = { 53, 359, 31, 95, 345, 52, 13, 58, 2, 78 };

        priorityQueueSort(numbers, 10);

        for (int i = 0; i < 10; i++) //should be in sorted order now
                cout << numbers[i] << endl;


        //Part 3:  Implement the "heap sort" algorithm using the STL built in priority queue.
        int size = 10; //replace with 10000000 to stress test and time
        int * bignums = new int[size];
        for (int i = 0; i < size; i++)
                bignums[i] = rand();

        clock_t start, end;

        start = clock();
        heapSort(bignums, size);
        end = clock();

        cout << "Heap sort took: " << end - start << " milleseconds." << endl;

        //comment out display for stress test
        for (int i = 0; i < size; i++)
                cout << bignums[i] << endl;

        return 0;
}

c++