Session 07-02 - Basic Probability Concepts

Section 07: Probability & Statistics

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Session 07-01

Test your understanding of Descriptive Statistics

Find the mean and median of: \(8, 12, 15, 9, 16, 12, 11\)
If the variance of a dataset is 16, what is the standard deviation?
A frequency table shows 15 out of 50 items are defective. What is the relative frequency of defective items?
What is the interquartile range if \(Q1 = 25\) and \(Q3 = 45\)?

Learning Objectives

What You’ll Master Today

Define sample spaces and events using proper notation
Apply probability axioms: \(0 \leq P(A) \leq 1\)
Use the complement rule: \(P(A') = 1 - P(A)\)
Apply the addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Distinguish between independent and mutually exclusive events
Solve probability problems in business contexts

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These concepts are fundamental for all probability calculations on the exam!

Part A: Sample Spaces and Events

Random Experiments

A random experiment is a process with uncertain outcomes.

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Examples:

Rolling a die
Selecting a product for quality control
Surveying a customer about satisfaction
Measuring daily sales

Sample Space

Definition: Sample Space (Ergebnismenge)

The sample space \(S\) (or \(\Omega\)) is the set of all possible outcomes of a random experiment.

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Examples:

Experiment	Sample Space
Coin flip	\(S = \{H, T\}\)
Die roll	\(S = \{1, 2, 3, 4, 5, 6\}\)
Two coin flips	\(S = \{HH, HT, TH, TT\}\)

Events

Definition: Event (Ereignis)

An event \(A\) is a subset of the sample space \(S\).

. . .

Example: Die roll with \(S = \{1, 2, 3, 4, 5, 6\}\)

Event A: “Rolling an even number” = \(\{2, 4, 6\}\)
Event B: “Rolling greater than 4” = \(\{5, 6\}\)
Event C: “Rolling a 7” = \(\emptyset\) (impossible event)
Event D: “Rolling a positive number” = \(S\) (certain event)

Set Operations on Events

Operation	Notation	Meaning
Union	\(A \cup B\)	A or B (or both)
Intersection	\(A \cap B\)	A and B
Complement	\(A'\) or \(\bar{A}\)	Not A

Part B: Probability Axioms

Definition of Probability

Kolmogorov Axioms

For any event \(A\):

\(P(A) \geq 0\) (non-negativity)
\(P(S) = 1\) (certainty)
For mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\)

. . .

Consequence: \(0 \leq P(A) \leq 1\) for all events \(A\)

Classical Probability

For equally likely outcomes:

\[P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{|A|}{|S|}\]

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Example: Rolling a fair die

\[P(\text{even}) = \frac{|\{2, 4, 6\}|}{|\{1,2,3,4,5,6\}|} = \frac{3}{6} = \frac{1}{2}\]

Complement Rule

Complement Rule (Gegenwahrscheinlichkeit)

\[P(A') = 1 - P(A)\]

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Example: If the probability of rain is 0.3, what is the probability of no rain?

\[P(\text{no rain}) = 1 - P(\text{rain}) = 1 - 0.3 = 0.7\]

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The complement rule is often useful when it’s easier to calculate what you don’t want!

Example: Using the Complement

A company knows that 5% of its products are defective.

Question: What is the probability that a randomly selected product is NOT defective?

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Solution: \[P(\text{defective}) = 0.05\] \[P(\text{not defective}) = 1 - 0.05 = 0.95\]

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Question: In a sample of 3 products, what’s the probability that at least one is defective?

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Solution: Use complement! \[P(\text{at least one defective}) = 1 - P(\text{none defective}) = 1 - (0.95)^3 \approx 0.143\]

Part C: Addition Rule

Union of Events

General Addition Rule (Additionssatz)

\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

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Why subtract \(P(A \cap B)\)?

Addition Rule Example

In a class of 100 students:

60 study mathematics
40 study economics
25 study both

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Question: What is the probability that a randomly selected student studies mathematics OR economics?

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Solution: \[P(M \cup E) = P(M) + P(E) - P(M \cap E)\] \[= 0.60 + 0.40 - 0.25 = 0.75\]

Part D: Mutually Exclusive Events

Mutually Exclusive (Disjoint) Events

Definition

Events A and B are mutually exclusive (disjunkt) if they cannot occur together:

\[A \cap B = \emptyset \quad \Rightarrow \quad P(A \cap B) = 0\]

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Examples:

Rolling a 3 and rolling a 5 on one die
A product being “good” and “defective”
Being in age group “18-25” and “26-35”

Special Addition Rule

For mutually exclusive events:

\[P(A \cup B) = P(A) + P(B)\]

. . .

Example: Rolling a die, find \(P(\text{1 or 6})\)

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Since rolling 1 and rolling 6 are mutually exclusive: \[P(1 \cup 6) = P(1) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\]

Break - 10 Minutes

Part E: Independent Events

Independence

Definition: Independent Events

Events A and B are independent (unabhängig) if the occurrence of one does not affect the probability of the other:

\[P(A \cap B) = P(A) \cdot P(B)\]

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Don’t confuse: - Mutually exclusive: Can’t happen together (\(P(A \cap B) = 0\)) - Independent: Don’t affect each other (\(P(A \cap B) = P(A) \cdot P(B)\))

Independence Example

Two machines work independently. Machine A has 95% reliability, Machine B has 90% reliability.

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Question: What is the probability both machines work?

\[P(A \cap B) = P(A) \cdot P(B) = 0.95 \times 0.90 = 0.855\]

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Question: What is the probability at least one machine fails?

\[P(\text{at least one fails}) = 1 - P(\text{both work}) = 1 - 0.855 = 0.145\]

Mutually Exclusive vs Independent

Property	Mutually Exclusive	Independent
\(P(A \cap B)\)	\(= 0\)	\(= P(A) \cdot P(B)\)
Can occur together?	No	Yes
Knowing A occurred…	…tells us B didn’t	…tells us nothing about B
Example	“Pass” vs “Fail”	Two separate coin flips

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If \(P(A) > 0\) and \(P(B) > 0\), then mutually exclusive events cannot be independent!

Part F: Business Applications

Quality Control Application

A factory produces items with:

3% have surface defects (event S)
2% have internal defects (event I)
0.5% have both defects

. . .

Find the probability that an item has:

At least one type of defect
A surface defect but no internal defect
Exactly one type of defect

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Solutions:

\(P(S \cup I) = P(S) + P(I) - P(S \cap I) = 0.03 + 0.02 - 0.005 = 0.045\)
\(P(S \cap I') = P(S) - P(S \cap I) = 0.03 - 0.005 = 0.025\)
\(P(\text{exactly one}) = P(S \cup I) - P(S \cap I) = 0.045 - 0.005 = 0.04\)

Market Research Application

In a survey of 500 consumers:

300 prefer Brand A
250 prefer organic products
150 prefer Brand A AND organic

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Question: Are “preferring Brand A” and “preferring organic” independent?

. . .

Check independence: \[P(A) \cdot P(\text{Org}) = \frac{300}{500} \times \frac{250}{500} = 0.6 \times 0.5 = 0.30\]

\[P(A \cap \text{Org}) = \frac{150}{500} = 0.30\]

. . .

Since \(P(A) \cdot P(\text{Org}) = P(A \cap \text{Org})\), the events are independent!

Guided Practice - 20 Minutes

Practice Problems

Work in pairs

Problem 1: A card is drawn from a standard 52-card deck. a) Find \(P(\text{Heart})\) b) Find \(P(\text{Face card})\) (J, Q, K) c) Find \(P(\text{Heart OR Face card})\)

Problem 2: In a company, 40% of employees are in sales, 30% are in engineering, and 10% are in both. Find: a) \(P(\text{Sales OR Engineering})\) b) \(P(\text{neither Sales nor Engineering})\)

Wrap-Up & Key Takeaways

Today’s Essential Concepts

Sample space \(S\): All possible outcomes
Event: Subset of the sample space
Probability axioms: \(0 \leq P(A) \leq 1\), \(P(S) = 1\)
Complement: \(P(A') = 1 - P(A)\)
Addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Mutually exclusive: \(P(A \cap B) = 0\)
Independent: \(P(A \cap B) = P(A) \cdot P(B)\)

Next Session Preview

Coming Up: Combinatorics

Fundamental counting principle
Permutations: arrangements where order matters
Combinations: selections where order doesn’t matter
Applications to probability calculations

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Homework

Complete Tasks 07-02: - Practice sample space identification - Apply probability rules - Solve business probability problems