1. Applied Statistical and Optimization Models
Topic 06-01: Basic Probability Concepts

2. Objectives Approaches to probability: Classical, Empirical, Subjective
Probability rules for events and their complements
Probability rules for unions of mutually and non-mutually exclusive events
Probability rules for intersections of dependent and independent events

3. Probability Approaches to Probability Classical Approach
A priori information is give
For example: Probability of rolling a “six” with a fair dice is P(“six”) = 1/6
Empirical Approach
No a priori information is given
Probability needs to be gauged by empirical work
For example: A survey of 100 randomly selected students revealed that 60% use the gym regularly. We therefore assume that there is a 60% chance that a randomly selected student goes to the gym regularly.
Subjective Approach
Empirical work not possible
Therefore only applicable to singular events
For example: Outcome of a boxing fight between two first time opponents.
Most of this chapter is concerned with the classical approach.

4. Probability Mutually and Non-Mutually Exclusive Events
Mutually exclusive: Rolling a “1” and a “six” in a single dice roll are mutually exclusive events.
Non mutually exclusive: Rolling an odd number and a number less than 5 are not mutually exclusive events.
A Venn diagram for two mutually exclusive events
A Venn diagram for two non-mutually exclusive events

5. Probability Probability of mutually exclusive events
Example: What’s the probability of getting a “5” or “6” in a single roll dice?
N(A) = Set of number of events for which we want to know the probability
A={5,6}
N(A) = 2
N(S) = Set of all possible events
S = {1,2,3,4,5,6}
N(S) = 6

6. Probability Probability of complements
The sum of the probability of event A to occur and event A not to occur (A’) must be equal to one: P(A)+P(A’)=1. A’ is the complement of A (Read: A prime is the complement of A).
Or, N(A)+N(A’)=N
Example: A bowl has 10 chocolate squares, 6 dark and 4 milk chocolate squares. In this case, the complementary event to drawing a dark chocolate square is drawing a milk chocolate square.
A Venn diagram for event A and its complement “not A” (A’).

7. Probability Unions and the Addition Rule
A ∪ B - read “A union B” (or “A OR B”)
P(A ∪ B) = P(A)+P(B) if A and B are mutually
exclusive
P(A ∪ B) = P(A)+P(B)–P(A ∩ B) if A and B are not mutually
where P(A ∩ B) reads “A intersection B” (or “A AND B”)

8. Probability
Example: Unions and the Addition Rule for mutually-exclusive events
Example for a union of mutually exclusive events: What’s the probability of drawing a “Jack” or a “Queen” in a random drawing from a deck of cards?
Answer: P(Jack ∪ Queen) = P(Jack)+P(Queen) = 4/52+4/52=8/52=2/13
Example: Unions and the Addition Rule for non mutually-exclusive events
Example for a union of non mutually exclusive events: What’s the probability of drawing a “Queen” or a “Heart” in a random drawing from a deck of cards?
Answer: P(Queen ∪ Heart) = P(Queen)+P(Heart)-P(Queen of Heart) = 4/52+13/52-1/52=16/52=4/13

9. Probability Intersections and the Multiplication Rules
A B “The intersection of A and B”
P(AB) = P(A)∙ P(B) if A and B are independent
P(AB) = P(A)∙ P(B|A) if B depends on A
 where P(B|A) is the probability of B given that A has occurred.
Test for Independence of two events
Two events are independent if
P(A)∙ P(B) = P(A)∙ P(B|A), which, of course simplifies to P(B) = P(B|A)   where P(B|A) is the probability of B given that A has occurred.

10. Probability
Example: Two independent events: The probability of rolling a “six” in the first and rolling a “six” in the second roll of a dice Answer: P(“Six””Six”) = P(“Six”)∙ P(“Six”)=1/6∙1/6=1/36
Example for two dependent events: A candy jar contains three dark chocolate squares and two milk chocolate squares. What is the probability that in two drawings without replacement both milk chocolate squares are selected?
Answer: P(“First is Milk””Second is Milk”) = P(“First is Milk”)∙P(“Second is Milk”|”First was Milk”) = 2/5 ∙ 1/4 = 2/20=1/10