1. An experiment consists of rolling six-sided dice twice.                                      (10)
   1. List the sample space for this experiment.
2. 2. Find the probability distribution for this experiment where x represents the number of even numbers in the 2 rolls.
3. 3. Find the mean of the probability distribution.
4. 4. Find the standard deviation of the probability distribution.
5. 5. Would it be unusual to get 2 even numbers? Why or why not? (show your work)

Recently, the bowling scores of a certain bowler were normally distributed with mean 202 and standard deviation 19.

​a) Find the probability that a score is from

185 to 205

​b) Find the probability that a score is from

165 to 175

​c) Find the probability that a score is greater than 200

​d) The best score is 299

Find the percentile that corresponds to this​ score, and explain what that number represents.

The average number of sugar in a generic brand of cereal is 660mg, and the standard deviation is 35mg. Assume the variable is normally distributed.

a.) if a single cereal is selected, find the probability that the sugar content will be more than 670mg.

b.) if a sample of 10 cereals is selected, find the probability that the mean of the sample will be larger than 670mg.

c.)Why is the probability for part (a) greater than for part (b)?

Rim is not a good student. Her exam strategy is to rely on luck for the next exam. The exam consists of 20 questions. Each question has 5 possible answers, only one of which is correct. Rim plans to guess the answer to each question. Answer the following questions:

1. Using the Binomial Formula, find the probability that Rim gets exactly 6 false answers?

2. What is the probability that Rim gets 5 or more correct answers?

3. What is the expected value of the number of correct answers? What is the standard deviation of the number of wrong answers?

Problem 2:

2. You are given different scenarios for the rates of returns for Intek and Zurick stocks.

1. What is the probability that returns will be lower or equal to 2% for Intek?

2. What is the probability that returns will be higher than 2% for Intek?

3. What is the probability that returns will be higher than 4% for Zurick?

4. Compute the expected return for each stock.

5. Compute the standard deviation of return for each stock

Which of the following is not a property of a binomial experiment?

Question 15 options:

Question 16

The probability distribution for the number of goals the Norse soccer team makes per game is given below;

Number of Goals Probability

0 0.05

1 0.15

2 0.35

3 0.30

4 0.15

Refer to the probabilities, what is the probability that in a given game the Norse will score 2 goals or more?

Question 16 options:

A uniform probability distribution is a continuous probability distribution where the probability that the random variable assumes a value in any interval of equal length is

Question 20 options:

1. In a company production line, the number of defective parts and their probabilities produced in an hour are shown in TABLE 1. Let x be the number of defective parts in an hour and K is P(X=K):

TABLE 1

1. How many defective parts are expected to be produced in an hour in the company’s production line?                                                                        

2. Compute the standard deviation of the defective parts produced in an hour by the company’s production line.                                              

3. Find the value of P(0 < X £ 3).                                                                         

2. Penny will play 2 games of badminton against Monica. Penny’s chances of winning each game is around 60 %. Let X denotes the number of Penny winning the game.
   1. Construct a probability distribution and cumulative probability distribution table for X.

2. Graph the probability distribution of X.

A city has just added 110 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions: (i) Each new recruit has a 0.3 probability of remaining with the police force until retirement. (ii) Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.35. (iii) The number of pensions that the city will provide on behalf of each new hire is independent of the number of pensions it will provide on behalf of any other new hire. Determine the probability that the city will provide at most 62 pensions to the 110 new hires and their husbands. Enter your answer as a number accurate to 4 decimal places.

Suppose that the number of printing mistakes on each page of a 200-page Mathematics book is independent of that on other pages. and it follows a Poisson distribution with mean 0.2.

(a) Find the probability that there is no printing mistake on page 23.

(b) Let page N be the first page which contains printing mistakes.

Find (i) the probability that N is less than or equal to 3,

(ii) the mean and variance of N.

(c) Let M be the number of pages which contain printing mistakes.Find the mean and variance of M.

(d) Suppose there is another 200-page Statistics book and there are 40 printing mistakes randomly and independently scattered through it.

Let Y be the number of printing mistakes on page 23.

(i) Which of the distributions - Bernoulli, binomial, geometric, Poisson, does Y follow?

(ii) Find the probability that there is no printing mistake on page 23.

C++

Text file contains numbers 92 87 65 49 92 100 100 100 82 75 64 55 100 98 -99

Modify your program from Exercise 1 so that it reads the information from the gradfile.txt file, reading until the end of file is encountered. You will need to first retrieve this file from the Lab 7 folder and place it in the same folder as your C++ source code. Run the program

#include <iostream>
using namespace std;

typedef int GradeType[100]; // declares a new data type:
float findAverage (const GradeType, int); // finds average of all grades
int findHighest (const GradeType, int); // finds highest of all grades
int findLowest (const GradeType, int); // finds lowest of all grades

int main()
{
GradeType grades; // the array holding the grades.
int numberOfGrades; // the number of grades read.
int pos; // index to the array.
float avgOfGrades; // contains the average of the grades.
int highestGrade; // contains the highest grade.
int lowestGrade; // contains the lowest grade.
// Read in the values into the array
pos = 0;
cout << "Please input a grade from 1 to 100, (or -99 to stop)" << endl;
cin >> grades[pos];

while (grades[pos] != -99)
{
pos++;
cin >> grades[pos];
}
numberOfGrades = pos--;
// call to the function to find average
avgOfGrades = findAverage(grades, numberOfGrades);
cout << endl << "The average of all the grades is " << avgOfGrades << endl;
  
highestGrade = findHighest(grades, numberOfGrades);
cout << endl << "The highest grade is " << highestGrade << endl;
  
lowestGrade = findLowest(grades, numberOfGrades);
cout << "The lowest grade is " << lowestGrade << endl;

return 0;
}
float findAverage (const GradeType array, int size)
{
float sum = 0; // holds the sum of all the numbers
for (int pos = 0; pos < size; pos++)
sum = sum + array[pos];
return (sum / size); //returns the average
}
int findHighest (const GradeType array, int size)
{
int highest = array[0];
for (int pos = 0; pos < size; pos++)
{
if ( highest < array[pos])
{
highest = array[pos];
}
}
return highest;
}
int findLowest (const GradeType array, int size)
{
int lowest = array[0];
for (int pos = 0; pos < size; pos++)
{
if ( lowest > array[pos])
{
lowest = array[pos];
}
}
return lowest;
}