Discrete Math 108

From mortality tables it has been determined that the probability of a 20-year old male non-smoker dying within the year is 0.0035. Suppose an insurance company wants to sell a $50,000 1-year life insurance policy to a 20-year old non-smoking male. What should they charge for the policy to break even?

$175 We need to calculate the expected value of the policy to the company. The company will pay out $50,000 if the 20-year old male dies within the year. The probability of this happening is 0.0035 and -50,000 x 0.0035 = -175. The probability of the 20-year old living is 1- 0.0035 = 0.9965. This is not worth anything to the company yet - until a premium is assigned to the policy. Thus the company expects to lose $175 per person on this policy. They should charge $175 for the policy to break even.

How many times do you visit a vertex when traveling either a Hamilton circuit or path?

1

What is a tautology?

A statement that is always true

The set of all elements that are under consideration for a particular problem or situation is known as:

A universal set

If you're placing pigeons in pigeonholes, and there are more pigeons than pigeonholes, which of the following must be TRUE?

At least one pigeonhole will contain more than one pigeon.

What is the binomial probability formula?

B(x; n, P) = C * P^x * (1 - P)^(n - x)

Which is the nearest neighbor method?

Choose the next closest city

When one event affects the outcome of another event, we call them:

Dependent Events

What do we call data that cannot be divided, which is distinct, and can only occur in certain values?

Discrete Data

Why are NAND and NOR gates considered to be functionally complete?

Each could be used exclusively to form any other gate operation.

What do we call the number of successful outcomes expected in an experiment?

Expected value

What kind of reasoning uses two unrelated events to prove a point?

False cause

How do we find the number of items in neither of two sets?

Find the number of items in the union of the two sets and subtract from the number of items in the universe.

Which of the following does a graph search NOT include?

Finding the chromatic number When you perform a graph search, you are essentially exploring the graph to find all the possible paths, vertices or edges in the graph. The chromatic number is the minimum number of colors required to color a graph such that no two adjacent vertices get the same color.

What kinds of graphs does Fleury's algorithm work for?

Graphs with an Euler path or circuit in it

Select the tautology from the statements below.

I either order pizza or I don't order pizza.

Let's say we have a variable value called ''rainy_night''. Now let's say you want to stay in your cozy house, but you don't want to go when it's warm and clear, just when it's a raining. You can use Boolean logic:

IF (rainy_night) THEN...

Which of the following is a conditional statement whose hypothesis is 'Joe has a red car' and whose conclusion is 'Billy gets to drive it'?

If Joe has a red car, then Billy gets to drive it

Which of the following is a conditional statement?

If a square is a rectangle, then a triangle is a shape.

What can be said about this graph?

It has an Euler Circuit

Which of the following is NOT true about a bijection?

It will be graphed in the Cartesian plane

The best Hamilton circuit in a weighted graph is the one where the total cost is what?

Least

A two-dimensional rectangular array of numbers representing a shape's coordinates is properly called a:

Matrix

A matrix B multiplied by its identity matrix is equal to what?

Matrix B

Which formula is equivalent to an OR gate?

NOR(NOR(X,Y),NOR(X,Y))

Find the solution to the recurrence relation f(n) = 4 f(n/8) + n2.

O(n^2)

Find the solution to the recurrence relation f(n) = 8 f(n/2) + n3.

O(n^3 logn)

Which of the following statements is FALSE for the Cartesian product of two sets A and B?

Only one element of set A relates to an element of set B.

Finding the minimum spanning tree by starting at a random node and adding the node with the lowest weight link is called _____.

Prim's Algorithm

The turning of a figure or object about a fixed point is called a:

Rotation

An example of a recurrence relation is _____.

Un = 3 Un-1

What is the term for the collection of elements from set A plus the elements in set B?

Union

When are vertices in a graph said to be adjacent?

When an edge passes between them.

Use De Morgan's Law to rewrite the negation of the statement: You get home by 10pm or you're grounded.

You get home by 10pm or you're not grounded. Let p = You get home by 10pm and q = You're grounded. The original statement is p OR q. The negation is NOT (p OR q), which is equivalent to (NOT p) AND (NOT q), by De Morgan's Law.

A spanning tree for n nodes uses _____ edges.

n-1

Design a logic circuit means

transfering a human-readable specification into a circuit diagram

Discrete math deals with problems concerning options that can be expressed as _____.

whole numbers

Knowing the generating function corresponding to the sequence of the natural numbers (except 0) in the first row on the image below, which would be the generating function for the sequence in the second row of the image below?

x^2 / (1 - x)^2

Which of these is an infinite sequence of all the non-zero even numbers beginning at number 2?

{2, 4, 6, 8, ...}

Which one of the following pairs of sets is equal?

A = {1,2,3} B = {3,1,2} Equal sets have the same elements in any order. {1, 2, 3} is therefore equal to {3, 1, 2} and to {3, 2, 1}.

Which of the following statements is FALSE?

A POSET is called a meet semilattice if every pair of elements has a "least upper bound' element.

What is an example of a logical fallacy?

Saying something is true or false based on emotions

16 Which of the following is the induction step in mathematical induction?

Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.

What is the degree in which the variables are different from the mean called?

Standard deviation

Which of these is always logically equivalent to the inverse?

The converse

Knowing the generation function for the sequence of the natural numbers except 0 as below in the first row, how would you describe the sequence for the generation function in the second row?

The even natural numbers, except 0 Knowing the scaling rule we see that the second generation function is the first one multiplied by 2. Therefore, the sequence for the second row generation function will be < 2, 4, 6, 8, ... > which are the even natural numbers, except 0.

If it is not possible to simplify a function using Karnaugh maps, this means that

The function might be simplified by another method or it is in its simplest form

What is the formula for binomial combinations?

nCx = n!/{(n - x)!x!}

The number of comparisons required to sort n items into a tree structure varies as _____.

nlog(n)

Work to sort, organize the data, or maintain the Tree Sort is referred to as _____.

overhead

Which element is 'minimal' in the following diagram?

p

What is the hypothesis in the following statement? 'If p is an even integer and q is an odd integer, then p + q is an odd integer.'

p is an even integer and q is an odd integer

What is NOT a step of the systematic procedure of designing a logic circuit?

use different wire colors to connect the components

Jon and Alex play a game where they each need to kick a ball into a net. They will each kick 11 balls, but Alex has a slight edge, with a 60% chance of scoring to Jon's 40%. What is the probability that Jon will score exactly 8 times?

2%

Calculate 4! (factorial)

24

A ball is dropped and begins bouncing. On the first bounce, the ball travels 3 feet. Each consecutive bounce is 1/8 the distance of the previous bounce. What is the total distance that the ball travels after it first hits the ground? Round to the nearest hundredth.

3.43 feet Finding the answer to this problem requires calculating the infinite value of the series, a1/(1 - r). Using that formula, we get: 3 / (1 - (1 / 8)) = 3 / (7 / 8) = 3 * (8 / 7) = 24 / 7 = 3.43 feet.

Jon and Alex play a game where they shoot water at a target in order to make a car move on a track. The person that gets their car over the finish line first wins. Alex and Jon decide to play the game 3 times. If they are evenly matched, then what is the probability that Jon will win 2 games?

38%

Given the recurrence f(n) = 4 f(n/2) + 1, how many sub-problems will a divide-and-conquer algorithm divide the original problem into, and what will be the size of those sub-problems?

4 sub-problems, each of size n/2

Sam is working on a computer programming project. He is creating a game that allows the player to shoot a ball into a basket in competition with the computer. The computer is set to make 60% of the shots on the medium level. What is the probability that the computer will make exactly 3 baskets out of 10 shots?

4% probability of success

Which group of numbers does not appear to be a sequence with a set pattern?

4, -13, 1, 5, 16, ...

Solve the expression 7P2 (P = permutation)

42 7P2 = 7! / (7 - 2)! = 5040 / 120 = 42 or 7 x 6 = 42

Calculate the probability of getting a single pair in a hand of poker. Round that probability to the nearest percent. Please note that a hand is 5 cards.

42% The total number of ways to get a single pair is given by 13 choose 1 times 4 choose 2 times 12 choose 3 times (4 choose 1)^3, since we're choosing from 13 possible values and 4 suits for the pair of cards and from 12 possible values and 4 possible suits (3 times) for the remaining three cards. This gives 1,098,240 possible ways to get a pair. Divide this by 52 choose 5 (the total number of hands) for a probability of about 42%.

A couple wants to plant some shrubs around a circular walkway. They have seven different shrubs. How many different ways can the shrubs be planted?

720 The formula for the number of permutations of n things arranged in a circle is (n-1)! Since n=7, then the number of ways the shrubs can be arranged is (7-1)! = 6! = 720 ways.

Find the sum 1 + 8 + 15 + 22 + 29 using the formula for an arithmetic series.

75

Given item A, which of the following would be the value of item B?

80

The rotation matrix for 270 degrees in the counterclockwise direction is the same one as the one for a:

90 degree clockwise rotation

If he eats a hamburger, then he will eat two bags of fries. Which of the following represents the hypothesis in the above conditional statement?

He eats a hamburger

The shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 90 degrees in the counterclockwise direction. What are the coordinates for A'B'C'?

(-1,1), (0,4), (1,-1)

he shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 90 degrees in the counterclockwise direction. What are the coordinates for A'B'C'?

(-1,1), (0,4), (1,-1)

You play a game with two six-sided dice. If you roll a sum of 6 or 8, you win $3. If you roll a sum of 11, you win $1, but for anything else, you lose $2. If you continue to play this game, what do you expect to win in the long run?

-$0.44 per roll There are 5 ways to get a sum of 6 when you roll two six-sided dice: 1 and 5, 2 and 4, 3 and 3, 5 and1, 4 and 2. There are also 5 ways to get a sum of 8 when you roll two six-sided dice: 2 and 6, 3 and 5, 4 and 4, 6 and 2, 5 and 3. So the probability of getting a sum of 6 or 8 is 10/36. There are 2 ways to get a sum of 11 when rolling two six-sided dice: 5 and 6, 6 and 5. So the probability of getting a sum of 11 is 2/36. The probability of anything else is 1 - (2/36) - (10/36) = 24/36. The expected value is calculated by multiplying the value of each event by its probability and then summing the results, so we get an expected value of: (3 x 10/36) + (1 x 2/36) + (-2 x 24/36) = -16/36, which is about -0.44.

If the domain of the function f(x) = x2 is all real numbers, what is the range of f?

0 to infinity

When ordering binary numbers in a Karnaugh Map, which of the following is the correct sequence?

00, 01 , 11, 10

The identity matrix can be likened to what number?

1

When finding an Euler path using Fleury's algorithm, if vertex 3 is your starting point, which vertex can you not go to?

1

Which two vertices can you connect to Eulerize this graph?

1 and 3

Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that girls will place 1st, 2nd, and 3rd in the marathon?

1/114 To solve this problem, you must look at the probabilities that the top three runners are girls and multiply them together, 5/20 x 4/19 x 3/18 = 1/114.

How many different values can a Boolean represent?

2

Annie writes the numbers 1 through 10 on note cards. She flips the cards over so she cannot see the number and selects three cards from the stack. What is the probability that she has selected the cards numbered 1, 2, and 3?

1/120 For this problem, the probability for drawing 1, 2, or 3 for the first card is 3/10 since all three of the cards (1, 2, or 3) are in the stack and can possibly be drawn. For the second card, we assume success in drawing one of those numbers for the first card so there are two of the desired numbers left and a total of 9 cards (2/9). For the third card we assume success in drawing two of the desired cards in the first and second picks so only one of the desired cards is left in the 8 remaining cards. (1/8). Multiply these together: 3/10 ∗ 2/9 ∗ 1/8 = 6/720 = 1/120

A newspaper company is selecting four houses to receive a free newspaper on your block. There are 10 houses on your block that are numbered 1-10. What is the probability that the four houses selected for a free newspaper will all be even numbered houses?

1/42 n is the number of items that is Number of houses = 10, that is n = 10 r is the number of items being chosen at a time = 4, that is r = 4 10C4 = 10!/4!(10-4)! = 210 is the number of possible outcomes. Number of favorable outcome (even number) = 5 Required probability = 5/210 = 1/42

How many edges does this graph have?

10

Consider an example with 100 students where 20 are taking discrete math, 30 are taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking discrete math and web design, 10 are taking Java and web design and 5 are taking all three classes. How many students are taking discrete math only?

11 First take the number of students taking discrete math and subtract the number of students taking both discrete and Java. Then subtract the number of students taking both discrete and web design: 20 - 6 - 8 = 6. This will subtract the number of students taking all three classes twice, so add the number of students taking all three classes (5) to get 11.

A vehicle license plate uses three numbers and three letters on each plate. The numbers are listed first and then the letters. The numbers used range from 0-9 and the letters used can be any letter of the 26 letters of the alphabet. On any given license plate, the letters can be repeated, but the numbers cannot be repeated. How many different plates are possible?

12,654,720 The correct answer is 10 * 9 * 8 * 26 * 26 * 26 = 12,654,720. There are 10 choices for the first number. However, there are only 9 choices for the second number and 8 choices for the third number because numbers cannot repeat. The letters can repeat, so there are 26 choices for each of the three letters on the license plate. If you chose 17,576,000, then you calculated 10 choices for each number. If you chose 11,232,000, then you did not allow the letters to repeat.

If 55 and 89 are two sequential terms of the Fibonacci sequence, what term would come next?

144 Each term in a Fibonacci sequence is found by adding the two previous terms, so in this case, the next term is 55 + 89 = 144.

What is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure?

Binomial Experiment

What do we call the probability of each success in a given binomial experiment?

Binomial distribution

How are logic circuits built?

By combining logic gates according to the function expression.

What is a probability formula that uses factorials to find the number of possible combinations of all the outcomes in an experiment?

Combination formula

Which statement best describes combinatorics?

Combinatorics is a fancy name for the study of counting.

What is a database NOT required to do?

Correct Data

What does the solution to a recurrence for a divide-and-conquer algorithm estimate?

The number of operations needed to solve the problem.

Jeanette asks 50 students in her math class if they are comfortable using technology. Thirty-eight say they are comfortable using technology. Calculate the theoretical and actual probability of this scenario.

Theoretical: 50% Actual: 76% Jeanette knows that there is a 50% probability for each individual student on answering either 'yes' or 'no'. Therefore, theoretically, 50% of the class should be comfortable with technology. Since Jeanette will be asking 50 students, her total number of trials is 50. After conducting her survey, Jeanette finds that 38 out of the 50 students feel comfortable with technology. She can make this number into both a ratio (38:50) and a percentage (76%) for her actual probability. Therefore, 76% of students are comfortable using technology in this class.

Calculate the probability of getting a straight in a hand of poker.

There is approximately a 1 in 255 chance of getting a straight in poker.

What does a computer use to store binary numbers?

Transistor switches in the open or closed position

Ally is trying to determine if an experiment is binomial. She is asking twenty people in front of a pet store if dogs' mouths are cleaner than a toilet. She asks the participants to answer true or false to her question. As far as she knows, the responses of the previous participants do not influence the responses of the other participants. Could this be considered binomial?

Yes, there are two possible outcomes, the outcomes are independent, and there is a fixed number of trials.

Which of these symbols is commonly used to enclose a matrix?

[ ]

Finding the minimum spanning tree by listing all possible spanning trees, totaling their weights, and selecting the tree with the lowest total weight is called _____.

explicit enumeration