1. MATH 2311
Chapter 2 Recap

2. Combinatorics
Permutations: These are ways of counting where each selection is treated differently than the others, or that order matters. Ordering an entire group of size n: n! Order r items of a group of size n: nPr

3. Combinatorics
Permutations with Repetition: Repetition in you selection (you are allowed to pick the same thing twice): nr Repetition in the items you are selecting from (duplicate items in your selection set): 𝑛! 𝑟!𝑠!𝑡!

4. Combinatorics
Combinations: These are selections of a smaller group from a larger group. There is no difference in treatment among selections and order does not matter. Selecting a group of size r from a group of size n: nCr

5. Examples:
You are selecting 5 children from a class of 30 to have different roles in a school play. You are arranging a family of 6 people in line for a group photo. You are trying to determine how many 4-digits PINs are possible. How many different arrangements can made from the letters: PQQRRS You are selecting an advisory committee of 5 from a group of 50.

6. Set Theory:
Set Union:
Set Compliment:
Set Intersection: A A B A B B

7. Set Theory:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 3, 4, 5} B = {1, 3, 5, 7, 9} C = {2, 4, 6, 8, 10} A U B (Ac ∩ B)c B ∩ C Ac (A U C)c

8. Draw A Venn Diagram to Represent the following:
In a group of 90 people surveyed: 36 are male, 35 have a driver’s license, and 40 work full time. 21 are males with a driver’s license work full time with a driver’s license. 15 are males that work full time. 9 are males that work full time with a driver’s license. How many do not fall into any category?

9. Basic Probability Rules:
To calculate the probability of an event occurring: 𝑃 𝐸 = 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑡𝑜𝑡𝑎𝑙
Rules:
Probabilities must be between 0 (impossible) to 1 (guaranteed).
The probability of all events occurring is 1.
The probability of an event not occurring is 1 – P(E)
The probability of one or the other disjoint events occurring is P(A) + P(B)
The probability of events A or B occurring is P(A) + P(B) – P(A and B)

10. Example:
You are selecting one card from a standard deck of 52 cards: What is the probability that the card is red or a club? What is the probability that the card is a queen or a black card? What is the probability that the card is an emperor? What is the probability that the card is not a spade? What is the probability that the card is a diamond, heart, spade or club?

11. Conditional Probability Rules:
Dependent events are events that influence one another (having a fender-bender if it is raining). Independent events do not influence one another (having a fender-bender and wearing a blue shirt). 𝑃 𝐴 𝐵 = 𝑃(𝐴∩𝐵) 𝑃(𝐵) P(A|B) is the probability that A occurs if B has already occurred.

12. Tests for Independence:
Events are independent if one of the following is true (if one is true, they will all be true): P(A|B) = P(A) P(A|~B) = P(A) P(A) x P(B) = P(A ∩ B)

13. Example:
You are conducting random inspections of vehicles using a parking garage. The probability that a vehicle entering the garage is either a truck or that it is blue is The probability that it is a truck is The probability that it is blue is What is the probability that it is blue and a truck? Given that the vehicle is a truck, what is the probability that it is blue? Are the events of (blue vehicle and truck) independent or not?