Section 1 – Sets
A = {1,2,3,4,5} B = {4,5,6,7,8} C = {7,8,9} D = the set of prime numbers
1) Find the following:
A U C
C’ ∩ (A U B)
(A ∩ B) ∩ D
Draw a Venn Diagram for one of these. Make sure to label which one you chose.
Bonus:
((A U B) U C) ∩ D
Section 2 – Symbolic Logic
p – “I studied hard for this test” q – “I made a good formula sheet” r – “I will pass”
2) Convert the following into symbolic logic, then construct and complete a truth table for your statement: “If I study hard for this test and make a good formula sheet, then I will pass.”
3) Create and complete a truth table for the following:
(p V q) ↔ (p ∧ r)
4) You have a hypothetical compound statement containing simple statements p,q,r and s. Construct a truth table for this statement. You only need columns for the simple statements. How many lines are in this truth table and explain why you need this number of lines.
Section 3 – Graph Theory
Consider the matrix:
1 2 3 4
1
0 1 0 1
2
1 0 1 0
3
1 1 0 0
4
0 1 1 0
5) Draw the graph this matrix describes. Is it directed or undirected? How many nodes and edges does it have? What is the maximum number of edges a graph of this type with this many nodes can have?
Section 4 – Counting and Probability
6) You have an urn containing 10 balls: 3 blue, 5 red, 2 yellow. We choose 3 balls, one at a time.
a. List all possible outcomes.
b. What is the probability of choosing red all three times (with replacement)
c. What is the probability of choosing red all three times (without replacement)
7) There are 10 rows of 3 seats per row. How many different ways are there for you and 2 friends to sit if you want to sit together in a row and all rows are empty?
8) You play a game where you flip a coin and roll a die. The following are the payouts for each result:
H & 1 - $1
H & 2 - $2
H & 3 - $3
H & 4 - $4
H & 5 - $5
H & 6 - $6
T & 1 - $2
T & 2 - $4
T & 3 - $6
T & 4 - $8
T & 5 - $10
T & 6 - $12
What is the expected value of payout for each turn you take in this game? What can you expect to win/lose if each turn costs $5 and you play 10 times?
9) You have data distributed N(10,1).
a. What percentile would a score of 11.5 have?
b. What score would you need to be in the 95th
percentile?
10) Find the mean and standard deviation for the following data:
5, 9, 10, 3, 2
Bonus:
There is a lottery game in which you pick 4 numbers (0-10). If you get 1 number correct, you win $10. If you get 2 numbers correct, you win $100. If you get 3 numbers correct, you win $1000. If you get all 4 numbers correct, you win $10000. Getting a number correct means your number matches the winning number in the correct place. What is the expected value of one lottery ticket? If each ticket costs $15 and you purchase 500 tickets, how much can you expect to win/lose?