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📚 Study Material: ISE 205 Probability and Statistics - Week 1 Introduction
Source Information: This study material is compiled from the provided lecture audio transcript and PDF/PowerPoint text for the ISE 205 Probability and Statistics course, Fall 2025-2026, taught by Dr. Burcu ÇARKLI YAVUZ.
1. Course Overview & Logistics
Welcome to the foundational concepts of Probability and Statistics! This course, ISE 205, aims to equip students with essential tools for understanding and analyzing data.
- Course Code & Title: ISE 205 Probability and Statistics
- Instructor: Dr. Burcu ÇARKLI YAVUZ (bcarkli@sakarya.edu.tr)
- Semester: 2025-2026 Fall
1.1. 14-Week Course Flow 🗓️
The course is structured to cover a comprehensive range of topics over 14 weeks:
- Introduction to Probability and Statistics
- Conditional Probability
- Random Variables and Their Types
- Random Variables and Their Properties
- Expectation and Variance
- Discrete Probability Distributions
- Continuous Probability Distributions
- Midterm Exam (Date to be announced on the department page)
- Descriptive Statistics
- Sampling, Statistical Estimation, Confidence Intervals
- Hypothesis Testing for Single Samples
- Hypothesis Testing for Two Samples
- Regression
- Analysis of Variance
1.2. Assessment Breakdown 📊
Your final success grade will be determined by:
- In-term Work: 50%
- Midterm Exam: 55% of in-term work
- Quiz 1: 15% of in-term work
- Quiz 2: 15% of in-term work
- Quiz 3: 15% of in-term work
- Final Exam: 50%
1.3. Recommended Resources 📖
For deeper understanding and reference, the following textbooks are recommended:
- Ross, Sheldon M. (2012). Introduction to Probability and Statistics for Engineers and Scientists. (4th ed., translated by Prof. Dr. Salih Çelebioğlu & Prof. Dr. Reşat Kasap). Nobel.
- Walpole, Ronald E. (2016). Probability and Statistics for Engineers and Scientists. (9th ed., translated by Prof. Dr. M. Akif BAKIR). Palme Yayıncılık.
- Montgomery, Douglas C. & Runger, George C. (2019). Applied Statistics and Probability for Engineers. (6th ed., translated by M. Terziler, T. Öner, E. Dalan Yıldırım, & Ş. Ayar Özbal). Palme Yayıncılık.
2. Fundamentals of Probability and Statistics
2.1. What is Probability? 💡
Probability is the numerical measure of the likelihood of an event occurring.
- It quantifies the frequency with which different outcomes can be observed in an experiment.
- It is the study of uncertainty in events.
- Expressed as a real number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
2.2. What is Statistics? 📈
Statistics is an applied branch of mathematics focused on the systematic collection, organization, analysis, interpretation, and presentation of data.
- It uses principles from probability theory to evaluate data numerically and graphically.
- Key activities include: data compilation, segmentation, summarization (tables, charts), experimental design, determining observation principles, analyzing sample information, interpreting findings, and generalizing results.
- ⚠️ Note: Core statistics topics will be covered in the weeks following the midterm exam.
2.3. Random Experiments
A random experiment is an experimental study where, even under constant environmental conditions, repeated observations yield different random values. The exact outcome cannot be precisely predicted beforehand.
2.4. Sample Space and Events
- Sample Space (S): The set of all possible outcomes of a random experiment.
- Elements can be verbal or numerical.
- Example 1: Single coin toss: S = {Heads, Tails} or S = {0, 1} (if Heads=0, Tails=1).
- Example 2: Two coin tosses: S = {HH, HT, TH, TT} or S = {0, 1, 2, 3} (e.g., number of heads).
- Event (A): Any subset of the sample space S. It represents a specific outcome or a group of outcomes.
2.5. Visualizing the Sample Space 🖼️
Several methods can be used to visualize the sample space:
- Listing: Explicitly writing out all possible outcomes.
- S = {Heads, Tails}
- S = {1, 2, 3, 4, 5, 6} (for a single die roll)
- Venn Diagram: Graphical representation showing relationships between events as circles within a rectangle (sample space).
- Contingency Table: A table displaying the frequency distribution of variables.
- Tree Diagram: Useful for sequential events, showing all possible paths and outcomes.
- Example: A table tennis match where the first player to win 3 sets wins. A tree diagram would map out all possible sequences of wins (e.g., AAA, AACA, CACC, etc.) until one player reaches 3 wins.
3. Core Probability Concepts & Counting Techniques
3.1. Calculating Probability ✅
The probability of an event A, denoted P(A), is calculated as:
- P(A) = n(A) / n(S)
n(A): Number of outcomes favorable to event A.n(S): Total number of possible outcomes in the sample space.
3.2. Stability Property
If an experiment is repeated many times under identical conditions, the relative frequency of an event (h(A) = n(A)/n) tends to stabilize and approach a constant value as the number of repetitions (n) increases. This constant value is the probability P(A).
- P(A) = lim (n→∞) [n(A) / n]
3.3. Basic Probability Axioms 📌
These fundamental rules govern probabilities:
- Range of Probability: For any event A, 0 ≤ P(A) ≤ 1.
- Probability of Sample Space: The probability of the entire sample space S (a certain event) is P(S) = 1.
- Probability of Impossible Event: The probability of an impossible event (denoted by the empty set ∅) is P(∅) = 0.
- Complement Rule: The probability that event A does not occur (its complement, A') is P(A') = 1 - P(A).
- Addition Rule for Any Two Events: For any two events A and B, the probability of A or B occurring is P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
- Addition Rule for Disjoint Events: If A and B are disjoint (mutually exclusive) events, meaning they cannot occur simultaneously (P(A ∩ B) = 0), then P(A ∪ B) = P(A) + P(B).
- Addition Rule for Three Events: For events A, B, and C: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)
3.4. Examples of Probability Axioms
- Example 1: Newspaper Readers
- In a group of 30 people: 17 read A, 13 read B, 5 read both.
- Only A: 17 - 5 = 12
- Only B: 13 - 5 = 8
- Neither: 30 - (12 + 8 + 5) = 5
- P(not reading any newspaper) = 5/30 = 1/6
- P(reading a newspaper) = (12+8+5)/30 = 25/30 = 5/6 (or 1 - 1/6 = 5/6)
- P(reading only A) = 12/30 = 2/5
- Example 2: Student Grades
- P(A)=0.2, P(B)=0.3, P(C)=0.25, P(D)=0.15, P(F)=0.1. F is failing.
- P(passing) = P(A) + P(B) + P(C) + P(D) = 0.2 + 0.3 + 0.25 + 0.15 = 0.9
- Alternatively, P(passing) = 1 - P(F) = 1 - 0.1 = 0.9
3.5. Counting Sample Points 🔢
To calculate probabilities, especially in complex scenarios, we need methods to count the number of possible outcomes.
3.5.1. Addition Rule (for counting)
- If event A can occur in
n1ways and event B can occur inn2ways, and A and B are disjoint, then A or B can occur inn1 + n2ways. - Example: Travel from Istanbul to Izmir: 2 train + 4 airline + 40 bus + 1 sea route = 47 different ways.
- Example: Connecting to a server: 3 VPN + 2 IP + 4 remote desktop = 9 different ways.
3.5.2. Multiplication Rule
- If event A can occur in
n1ways, and for each of these ways, event B can occur inn2ways, then the sequence of A and B can occur inn1 * n2ways. - Example 1: Building a computer: 2 processor * 3 motherboard * 4 hard disk * 5 memory = 120 different computers.
- Example 2: Computer lab environments: 3 OS * 4 browsers = 12 different usage environments.
- Example 3: Forming 3-digit numbers using 0,1,2,3,4,5 (digits can repeat):
- Hundreds digit: 5 options (cannot be 0)
- Tens digit: 6 options
- Units digit: 6 options
- Total: 5 * 6 * 6 = 180 different 3-digit numbers.
- Example 4: Forming 4-digit even numbers using 0,1,2,3,4,5 (digits cannot repeat):
- Case 1: Units digit is 0 (1 option). Remaining 3 digits from 5 options: 5 * 4 * 3 * 1 = 60.
- Case 2: Units digit is 2 or 4 (2 options).
- Thousands digit: 4 options (cannot be 0 or the chosen units digit).
- Hundreds digit: 4 options.
- Tens digit: 3 options.
- Total: 4 * 4 * 3 * 2 = 96.
- Total 4-digit even numbers: 60 + 96 = 156.
3.5.3. Permutation
- An arrangement of objects where the order matters.
- The number of permutations of
robjects chosen fromndistinct objects is: P(n, r) = n! / (n - r)! - If all
nobjects are arranged: P(n, n) = n! - Circular Permutation: For
nobjects arranged in a circle, the number of permutations is (n - 1)!. - Permutation with Repetition: If there are
n1identical objects of type 1,n2of type 2, ...,nkof type k, the number of distinct permutations is n! / (n1! * n2! * ... * nk!). - Example 1: 8 contestants, first three places: P(8, 3) = 8! / (8-3)! = 8 * 7 * 6 = 336 ways.
- Example 2: 4-digit numbers with distinct digits from 2,3,5,6,7,9: P(6, 4) = 6! / (6-4)! = 6 * 5 * 4 * 3 = 360 numbers.
3.5.4. Combination
- A selection of objects where the order does not matter.
- The number of combinations of
robjects chosen fromndistinct objects is: C(n, r) = n! / (r! * (n - r)!) - Example 1: Selecting a 3-person commission from 5 people: C(5, 3) = 5! / (3! * 2!) = 10 ways.
- Example 2: Forming a committee with 2 men and 1 woman from 10 men and 5 women:
- Choose 2 men from 10: C(10, 2) = 10! / (2! * 8!) = 45
- Choose 1 woman from 5: C(5, 1) = 5! / (1! * 4!) = 5
- Total ways (using Multiplication Rule): 45 * 5 = 225 ways.
- Example 3: Probability of a commission having a majority of business students (from 10 business, 8 engineering, 5-person commission):
- Total ways to form a 5-person commission from 18 students: C(18, 5) = 8568
- Ways with majority business (3B 2E, 4B 1E, 5B 0E):
- C(10,3)*C(8,2) = 120 * 28 = 3360
- C(10,4)*C(8,1) = 210 * 8 = 1680
- C(10,5)*C(8,0) = 252 * 1 = 252
- Total favorable ways = 3360 + 1680 + 252 = 5292
- Probability = 5292 / 8568 ≈ 0.6176
