Part I: Problems about §11.5

1. Review the group exercises and assigned homework problems from this section . . .

 
2. This series of exercises gives additional practice similar to a series of exercises that we did in class.

  An airline is going to book n passengers for a flight, where n is a (possibly unknown) number. Let p be the probability that that a given passenger will actually board the flight.
 
1. Determine the expected number  of passengers that actually board the flight:

(i) if p = .85 (thus 15% no-show), then 
(ii) if p = .95 (thus 5% no-show), then 
(Your answer will be an expression that involves n.)
 
2. Determine the standard deviation  in the number of passengers that actually board the flight:

(i) if p = .85, then 
(ii) if p = .95, then 
(Your answer here also will be an expression that involves n.)
 
3. Find the z value (from the table in the notes) such that 85% of the area under the normal curve is to the left of that value.

Suggestions: (1) find the value of A(z) by solving the equation .5 + A(z) = .85
   (2) determine which column of the table in which to locate this value!!
 
4. Suppose that the airplane seats 200 passengers. Write the equation which expresses the condition that the airline can book n passengers and will not have to "bump" passengers 85% of the time:

(i) if p = .85
(ii) if p = .95
 
5. Simplify and solve your equations, and determine how many tickets the airline can sell.

Suggestions: Starting from the same equation as in Exercise 11.5.9 in the notes, you can get an equation that involves n and . You can turn it into a quadratic equation by using the auxiliary unknown . (Note that this gives .) Use the quadratic formula to solve this version of the equation.

Part II: Problems about chapter 12

1. Review the group exercises and assigned homework problems from this chapter . . .

 
2. Find the multiplicative inverses of the nonzero elements of F₇, the system of integers mod 7.

(In other words, find the solutions of the equations ax= 1 in F₇, for a = 1, 2, 3, 4, 5, 6.)
  Here, it may be useful to review section 6.3, from the chapter on modular arithmetic.
 
3. This set of exercises refers to the multiplication table for a field with 4 elements that was constructed in class on Wednesday.


To have a notation for this system, we will call this field F₄. Note, however, that it is not the system of integers mod 4.
As usual, the system of integers mod 2 will be denoted F₂. As explained in class, F₄ can be considered as an extension of F₂, so that identities such as 1 + 1 = 0 which hold in F₂ will continue to hold in F₄,

1. Use your addition and multiplication tables to calculate the value (in F₄) of x² + x + 1 for:

  (i) x = 0 (ii) x = 1 (iii) x =  (iv) x = 
 
2. What are the roots (in F₄) of the equation x² + x + 1 = 0?

Suggestion: use the results of part (a) to answer this.
 
3. Fill in the following chart with coefficients from F₂ (integers mod 2).

[The mathematical point here is that every element of F₄ turns out to be of the form , where a and b are elements of F₂.]




Comment: The entries here will just be zeros and ones!
 
4. Verify that 

Suggestion: This actually follows from one of the previous parts of this exercise.
Just remember that -1 = +1 in F₂ and therefore also in F₄,
 
5. Let a, b, c, d be elements of F₂. Calculate the following expression:
    
   
   Use the result of d) to simplify the answer (so thatjust appears to the 1^st power). Your answer should be of the following form:

   

6. Use the multiplication table to calculate:

 




Comment: The entries here should be elements of F₄,
 
7. Do you see a pattern in the above?