Essential of Probability
TUGAS WEEK 10
1 Introduction
Definition of Probability
Probability is a fundamental branch of mathematics and statistics that focuses on measuring and analyzing uncertainty. Etymologically, the word “probability” comes from the Latin “probabilitas” meaning “provable” or “credible”. In modern context, probability is defined as a numerical measure that describes how likely an event is to occur, with values ranging from 0 (impossible event) to 1 (certain event).
Probability is not merely a theoretical concept, but has very broad and deep applications in everyday life. Every day, without realizing it, we make decisions based on probability considerations. When we see cloudy skies and decide to bring an umbrella, we are actually estimating the probability of rain. When a doctor prescribes medication based on therapy success rates, they are using probability concepts to maximize patient recovery chances.
In the world of business and economics, probability becomes the foundation for strategic decision-making. Insurance companies calculate premiums based on claim probabilities, investors analyze portfolio risks using probability theory, and technology companies develop machine learning algorithms based on probabilistic principles. Even in social sciences, researchers use probability to analyze societal behavior patterns and predict demographic trends.
Historical Context
The history of probability began in the 16th century when Italian mathematician Gerolamo Cardano started analyzing dice games systematically. However, significant development occurred in the 17th century through the famous correspondence between Blaise Pascal and Pierre de Fermat about the “problem of points” - a problem of fair stake division in an interrupted game. Their work laid the foundation for modern probability theory.
In the 18th century, Jakob Bernoulli introduced the “Law of Large Numbers” which explains how relative frequency approaches theoretical probability as the number of trials increases. Subsequently, Pierre-Simon Laplace developed probability theory into a systematic mathematical discipline and introduced the concept of normal distribution. In the 20th century, Andrey Kolmogorov provided a rigorous axiomatic foundation for probability theory, making it a rigorous branch of mathematics.
This research is designed with comprehensive objectives to provide in-depth understanding of fundamental probability concepts. Starting from basic definitions and essential terminology, we will explore various types of events and relationships between events. This material will also discuss in detail operations in set theory relevant to probability, such as union, intersection, and complement.
Special focus is given to the binomial distribution, one of the most important and widely applied discrete probability distributions. Through a combination of mathematical theory, applicative examples, and engaging data visualizations, this research aims to equip readers with the ability not only to understand concepts theoretically, but also to apply them in practical situations.
- Primary Learning Objectives: Building comprehensive understanding of probability theory from basic level to advanced applications, enabling students to apply probabilistic reasoning in real-world scenarios
- Methodology: Integrative approach combining conceptual explanations, mathematical derivations, descriptive statistical analysis, and interactive data visualization to facilitate deep learning
- Material Coverage: Six core topics covering basic probability concepts, event classification (independent and dependent), set operations (union of events), special event properties (mutually exclusive and exhaustive), binomial experiments, and binomial distribution
- Pedagogical Approach: Each concept is explained through three stages - formal definition, concrete examples from real life, and visual representation that facilitates concept internalization
- Practical Relevance: Material is designed to demonstrate direct applications in various fields such as risk analysis, quality control, medical research, and data science
Important Note: A solid understanding of the concepts in this document will serve as an important foundation for advanced statistical topics such as statistical inference, hypothesis testing, regression analysis, and machine learning. Each section is designed to build understanding progressively and cumulatively, ensuring that students develop both theoretical knowledge and practical skills in probability and statistics.
2 Basic Concept of Probability
Formal Definition of Probability
Probability is a numerical function that maps every event in the sample space to a real number between 0 and 1. Mathematically, for every event A in sample space S, probability P(A) satisfies three Kolmogorov axioms:
- Non-negativity: P(A) ≥ 0 for all events A
- Normalization: P(S) = 1, where S is the sample space
- Additivity: For mutually exclusive events, P(A₁ ∪ A₂) = P(A₁) + P(A₂)
Classical Probability Formula:
P(A) = n(A) /
n(S)
where:
n(A) = number of favorable outcomes
n(S) =
total number of outcomes in sample space
Fundamental Terminology
Key Concepts
1. Sample Space (S): The set of all possible outcomes that can occur in a random experiment. The sample space is exhaustive (covers all possibilities) and mutually exclusive (each outcome is distinct).
2. Event (A, B, C, …): A subset or portion of the sample space that we are interested in or want to calculate the probability of. An event can consist of one outcome (simple event) or several outcomes (compound event).
3. Outcome: An individual result or single element from the sample space. Each time an experiment is conducted, exactly one outcome will occur.
4. Random Experiment: A process or activity that can produce different outcomes, where the specific outcome that will occur cannot be predicted with certainty before the experiment is conducted.
Sample Space: S = {1, 2, 3, 4, 5, 6}
Event - Even Number: A = {2, 4, 6}, P(A) = 3/6 = 0.5 or 50%
Event - Prime Number: B = {2, 3, 5}, P(B) = 3/6 = 0.5 or 50%
Event - Number Greater than 4: C = {5, 6}, P(C) = 2/6 ≈ 0.333 or 33.33%
Event - Number Less than 7: D = {1, 2, 3, 4, 5, 6} = S, P(D) = 6/6 = 1.0 or 100% (certain event)
Event - Number Equals 7: E = ∅ (empty set), P(E) = 0/6 = 0 or 0% (impossible event)
Each outcome has equal probability of 1/6 ≈ 0.167 (Uniform Distribution)
3 Independent and Dependent Events
Independent Events
Two events A and B are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically:
P(A ∩ B) = P(A) × P(B)
Or equivalently: P(A|B) = P(A) and P(B|A) = P(B)
Dependent Events
Events A and B are dependent if the occurrence of one event affects the probability of the other event:
P(A ∩ B) = P(A) × P(B|A)
Or: P(A ∩ B) = P(B) × P(A|B)
Multiplication Rule
For Independent Events:
P(A ∩ B) = P(A) ×
P(B)
For Dependent Events:
P(A ∩ B) = P(A)
× P(B|A)
P(A ∩ B) = P(B) × P(A|B)
Drawing a card, recording it, returning it to the deck, then drawing again:
• P(first card is Heart) = 13/52 = 0.25
• P(second card is Heart | first is Heart) = 13/52 = 0.25
• P(both Hearts) = 0.25 × 0.25 = 0.0625 or 6.25%
Analysis: The probability remains constant because the first card is replaced, making the events independent.
Drawing the first card and not returning it to the deck:
• P(first card is Heart) = 13/52 = 0.25
• P(second card is Heart | first is Heart) = 12/51 ≈ 0.235
• P(both Hearts) = (13/52) × (12/51) ≈ 0.0588 or 5.88%
Analysis: The probability changes because removing one Heart card affects the composition of the remaining deck.
Notice: Without replacement, the probability changes for the second draw, showing dependence
Key Insights on Independence
Testing for Independence:
- Check if P(A ∩ B) = P(A) × P(B)
- Or check if P(A|B) = P(A)
- If either condition holds, events are independent
Common Misconceptions:
- Independent ≠ Mutually Exclusive: Independent events can occur together; mutually exclusive events cannot
- Zero correlation ≠ Independence: Events can have zero correlation but still be dependent
4 Union of Events
Addition Rule
General Addition Rule:
P(A ∪ B) = P(A) +
P(B) - P(A ∩ B)
For Mutually Exclusive
Events:
P(A ∪ B) = P(A) + P(B)
(since P(A ∩ B) = 0)
Understanding Union
The union of events A and B, denoted as A ∪ B, represents the event that at least one of the events occurs. In other words, A ∪ B occurs if A occurs, or B occurs, or both occur.
Why subtract P(A ∩ B)? When we add P(A) and P(B), we count the intersection twice. Subtracting P(A ∩ B) once corrects this double-counting.
From a standard deck of 52 cards:
• Event A = Drawing a Heart (13 cards)
• Event B = Drawing a King (4 cards)
• Intersection = King of Hearts (1 card)
Calculation:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 13/52 + 4/52 - 1/52 = 16/52 ≈ 0.308 or 30.8%
Rolling a die:
• Event A = Getting an even number {2, 4, 6}
• Event B = Getting an odd number {1, 3, 5}
• P(A ∩ B) = 0 (cannot be both even and odd)
Calculation:
P(A ∪ B) = P(A) + P(B) = 3/6 + 3/6 = 6/6 = 1.0
This equals 1 because A and B are mutually exclusive and exhaustive.
Extension to Three or More Events
For Three Events:
P(A ∪ B ∪ C) = P(A) + P(B) +
P(C)
- P(A ∩ B) - P(A ∩ C) - P(B ∩ C)
+ P(A ∩ B ∩ C)
Important Properties of Union
- Commutative: A ∪ B = B ∪ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Identity: A ∪ ∅ = A
- Idempotent: A ∪ A = A
- Absorption: If A ⊆ B, then A ∪ B = B
5 Mutually Exclusive and Exhaustive Events
Mutually Exclusive Events
Two or more events are said to be mutually exclusive (or disjoint) if they cannot occur simultaneously. Mathematically:
A ∩ B = ∅ and therefore P(A ∩ B) = 0
For multiple events A₁, A₂, …, Aₙ to be mutually exclusive, every pair must satisfy: Aᵢ ∩ Aⱼ = ∅ for all i ≠ j
Exhaustive Events
Events are called exhaustive if their union covers the entire sample space. At least one of the events must occur:
A₁ ∪ A₂ ∪ … ∪ Aₙ = S
For mutually exclusive and exhaustive events: Σ P(Aᵢ) = 1
Partition of Sample Space
When events are both mutually exclusive AND exhaustive, they form a partition of the sample space. This means:
- Every outcome belongs to exactly one event
- No outcome is left out
- No outcome belongs to multiple events
Partition the die outcomes into two events:
• Event A = {1, 2, 3} “Small numbers”, P(A) = 3/6 = 0.5
• Event B = {4, 5, 6} “Large numbers”, P(B) = 3/6 = 0.5
Verification:
✓ Mutually Exclusive: A ∩ B = ∅
✓ Exhaustive: A ∪ B = S
✓ P(A) + P(B) = 0.5 + 0.5 = 1.0
Patient diagnosis results (mutually exclusive & exhaustive):
• Positive diagnosis: P = 0.15
• Negative diagnosis: P = 0.82
• Inconclusive: P = 0.03
Total: 0.15 + 0.82 + 0.03 = 1.00 ✓
Every patient falls into exactly one category.
From a die roll:
• Event A = {1, 2} “Very Low”, P(A) = 2/6 ≈ 0.333
• Event B = {5, 6} “High”, P(B) = 2/6 ≈ 0.333
Analysis:
✓ Mutually Exclusive: A ∩ B = ∅
✗ NOT Exhaustive: {3, 4} is not covered!
P(A ∪ B) = 0.667 ≠ 1.0
Law of Total Probability
Law of Total Probability:
If A₁, A₂, …, Aₙ
form a partition of S (ME & Exhaustive),
then for any event
B:
P(B) = P(B|A₁)×P(A₁) + P(B|A₂)×P(A₂) + … +
P(B|Aₙ)×P(Aₙ)
P(B) = Σ P(B|Aᵢ)×P(Aᵢ)
Key Insights
Why Partitions are Powerful:
- Simplification: Breaking complex sample spaces into manageable pieces
- No double-counting: Each outcome belongs to exactly one partition element
- Complete accounting: No outcomes left out
- Enables conditioning: Foundation for Law of Total Probability and Bayes’ Theorem
Common Misconceptions:
- ME ≠ Independent: Mutually exclusive events (with non-zero probabilities) are ALWAYS dependent
- Complement is special case: A and A’ form the simplest ME & Exhaustive pair, so P(A) + P(A’) = 1
- Exhaustive doesn’t imply ME: Events can cover entire sample space but still overlap
6 Binomial Experiment and Formula
Formal Definition of Binomial Experiment
A binomial experiment is a special type of random experiment that satisfies four strict conditions (often abbreviated as BINS):
-
Binary outcomes (B): Each trial has only two mutually
exclusive possible outcomes:
- “Success” (S) with probability p
- “Failure” (F) with probability q = 1 - p
- Independent trials (I): The outcome of one trial does not affect the outcome of other trials; P(success on trial i | outcomes of other trials) = p for all i
- Number of trials is fixed (N): The number of trials n is determined before the experiment begins and does not change
- Same probability (S): The probability of success p is constant for each trial; it does not change throughout the experiment
If all four conditions are met, the random variable X = “number of successes in n trials” follows a binomial distribution.
Derivation of Binomial Formula
Scenario: n = 3 trials, calculate P(X = 2 successes)
Possible sequences with exactly 2 successes:
- SSF - Success, Success, Failure
- SFS - Success, Failure, Success
- FSS - Failure, Success, Success
Probability of each specific sequence:
- P(SSF) = p × p × (1-p) = p²(1-p)
- P(SFS) = p × (1-p) × p = p²(1-p)
- P(FSS) = (1-p) × p × p = p²(1-p)
All sequences have the same probability: p²(1-p)
Number of sequences with 2 successes: C(3,2) = 3!/(2!×1!) = 3
Total probability: P(X = 2) = 3 × p²(1-p)
Binomial Probability Mass Function:
P(X = k)
= C(n, k) × p^k × (1-p)^(n-k)
P(X = k) = [n!/(k!(n-k)!)] × p^k ×
q^(n-k)
where:
• n = number of trials (fixed)
• k =
number of successes (0 ≤ k ≤ n)
• p = probability of success on each
trial
• q = 1-p = probability of failure
• C(n,k) = binomial
coefficient = “n choose k”
Properties:
• Σ
P(X = k) = 1 for k = 0 to n (valid PMF)
• Mode occurs at ⌊(n+1)p⌋
(most likely outcome)
• Mean: E(X) = np
• Variance: Var(X) =
np(1-p)
Components of the Formula
Understanding Each Component
1. C(n,k) - Binomial Coefficient:
Counts the number of different sequences of n trials that result in exactly k successes. This is purely combinatorial.
2. p^k - Probability of Successes:
The probability of any specific sequence with k successes. Raised to the k-th power because we need k independent success events.
3. (1-p)^(n-k) - Probability of Failures:
The probability of any specific sequence with (n-k) failures that necessarily accompany k successes in n trials.
4. Product - Final Probability:
Multiply all three components: how many ways × probability of successes × probability of failures
Flip a fair coin 5 times, calculate P(exactly 3 heads):
• n = 5 trials (fixed)
• p = 0.5 (probability of heads)
• k = 3 (desired successes)
• C(5,3) = 5!/(3!×2!) = 10 different sequences
Calculation:
P(X = 3) = 10 × (0.5)³ × (0.5)²
P(X = 3) = 10 × 0.125 × 0.25
P(X = 3) = 0.3125 or 31.25%
A drug has 70% success rate, treat 15 patients:
• n = 15 independent patients
• p = 0.7 (success probability)
• Expected successes: μ = np = 15×0.7 = 10.5 patients
Calculations:
P(exactly 10 successes) = C(15,10) × (0.7)¹⁰ × (0.3)⁵
= 3003 × 0.0282 × 0.00243 ≈ 0.2061 or 20.61%
P(at least 12 successes) = P(12) + P(13) + P(14) + P(15) ≈ 0.2969 or 29.69%
Defect rate is 5%, inspect 20 items:
• n = 20 items
• p = 0.05 (defect probability)
Calculations:
P(0 defects) = C(20,0) × (0.05)⁰ × (0.95)²⁰ ≈ 0.3585 or 35.85%
P(1 defect) = C(20,1) × (0.05)¹ × (0.95)¹⁹ ≈ 0.3774 or 37.74%
P(at most 1 defect) = P(0) + P(1) ≈ 0.7359 or 73.59%
Expected defects: μ = 20×0.05 = 1 defect
The distribution peaks around k=6 (the expected value: np = 10×0.6 = 6)
When to Use Binomial Model
Appropriate When:
- ✓ Binary outcomes (success/failure, yes/no, pass/fail)
- ✓ Fixed number of trials known in advance
- ✓ Trials are independent (outcome of one doesn’t affect others)
- ✓ Probability constant across trials
NOT Appropriate When:
- ✗ More than two possible outcomes per trial
- ✗ Number of trials not fixed (keep going until success)
- ✗ Trials are dependent (sampling without replacement from small population)
- ✗ Probability changes across trials (learning effects, fatigue, etc.)
Approximations:
- Normal Approximation: When n large AND np > 5 AND n(1-p) > 5
- Poisson Approximation: When n large, p small, BUT np moderate (np < 10)
7 Binomial Distribution
Definition of Binomial Distribution
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success on each trial). The binomial random variable X represents the count of successes, where X can take values from 0 to n.
Mathematically, X ~ Binomial(n, p), and the probability mass function (PMF) is given by:
P(X = k) = C(n, k) × p^k × (1-p)^(n-k)
for k = 0, 1, 2, …, n.
The binomial distribution arises naturally in scenarios involving repeated independent trials with binary outcomes, such as coin flips, quality control inspections, medical trials, and opinion polls.
Parameters and Characteristics
Key Parameters
1. Number of Trials (n): A positive integer representing the fixed number of independent trials. This parameter determines the range of possible values for X (0 to n).
2. Probability of Success (p): A real number between 0 and 1, where p is the probability of success on each individual trial, and q = 1 - p is the probability of failure.
3. Support: The set of possible values for X is {0, 1, 2, …, n}. Each value k represents exactly k successes in n trials.
4. Shape Characteristics:
- When p = 0.5, the distribution is symmetric.
- When p < 0.5, the distribution is right-skewed.
- When p > 0.5, the distribution is left-skewed.
- As n increases, the distribution becomes more bell-shaped (approaching normal distribution).
Mean, Variance, and Standard Deviation
Mean (Expected Value):
E[X] = μ = n × p
Variance:
Var(X) = σ² = n × p × (1-p)
Standard Deviation:
σ = √(n × p × (1-p))
Mode:
The most likely value is the integer closest
to (n+1)p
Skewness:
γ₁ = [1 - 2p] / √[n p
(1-p)]
Kurtosis:
γ₂ = [1 - 6p(1-p)] / [n p
(1-p)]
Consider a binomial distribution with n = 20 trials and p = 0.3 (success probability).
Mean: E[X] = 20 × 0.3 = 6 successes
Variance: Var(X) = 20 × 0.3 × 0.7 = 4.2
Standard Deviation: σ = √4.2 ≈ 2.05
Interpretation: On average, we expect 6 successes, but the actual number can vary with a standard deviation of about 2.05 around this mean.
Cumulative Distribution Function (CDF)
Cumulative Probabilities
The cumulative distribution function (CDF) gives the probability that the random variable X is less than or equal to a specific value k:
F(k) = P(X ≤ k) = Σ P(X = i) for i = 0 to k
This is useful for calculating probabilities of ranges, such as “at most k successes” or “at least k successes”.
Complementary CDF: P(X ≥ k) = 1 - P(X ≤ k-1)
Range Probabilities: P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a-1)
Using the binomial distribution with n = 10, p = 0.4:
• P(X ≤ 3) = P(0) + P(1) + P(2) + P(3)
• P(X ≥ 7) = P(7) + P(8) + P(9) + P(10) = 1 - P(X ≤ 6)
• P(4 ≤ X ≤ 6) = P(X ≤ 6) - P(X ≤ 3)
Calculated Values (approximate):
• P(X ≤ 3) ≈ 0.382
• P(X ≥ 7) ≈ 0.167
• P(4 ≤ X ≤ 6) ≈ 0.451
Applications and Real-World Examples
A factory produces light bulbs with a defect rate of 5%. If we inspect 50 bulbs, what is the probability of finding exactly 3 defective bulbs?
• n = 50, p = 0.05, k = 3
• P(X = 3) = C(50, 3) × (0.05)^3 × (0.95)^47
• Calculation: C(50,3) ≈ 19600, (0.05)^3 ≈ 0.000125, (0.95)^47 ≈ 0.076
• P(X = 3) ≈ 19600 × 0.000125 × 0.076 ≈ 0.185 or 18.5%
Further Analysis:
• Expected defects: μ = 50 × 0.05 = 2.5
• P(at most 2 defects) = P(X ≤ 2) ≈ 0.749
• P(more than 5 defects) = P(X ≥ 6) ≈ 0.109
A new drug has a 75% success rate in treating a condition. In a clinical trial with 12 patients, calculate the probability that at least 9 patients respond positively.
• n = 12, p = 0.75, k ≥ 9
• P(X ≥ 9) = P(9) + P(10) + P(11) + P(12)
• Using binomial formula or tables:
• P(X = 9) = C(12,9) × (0.75)^9 × (0.25)^3 ≈ 0.258
• P(X = 10) ≈ 0.232, P(X = 11) ≈ 0.103, P(X = 12) ≈ 0.032
• Total P(X ≥ 9) ≈ 0.625 or 62.5%
Expected successes: μ = 12 × 0.75 = 9
In a survey of 100 voters, 60% support a candidate. What’s the probability that between 55 and 65 supporters are found?
• n = 100, p = 0.60
• P(55 ≤ X ≤ 65) = P(X ≤ 65) - P(X ≤ 54)
• Using normal approximation (since n is large): μ = 60, σ = √(100×0.6×0.4) = √24 ≈ 4.9
• P(55 ≤ X ≤ 65) ≈ P(54.5 ≤ X ≤ 65.5) ≈ Φ(1.12) - Φ(-1.12) ≈ 0.868 - 0.132 = 0.736
Exact calculation would be more precise but computationally intensive.
Visualizations and Shape Analysis
Notice how the shape changes: right-skewed for p<0.5, symmetric for p=0.5, left-skewed for p>0.5
Relationships with Other Distributions
Connections to Other Probability Distributions
1. Bernoulli Distribution: The binomial distribution is the sum of n independent Bernoulli trials. When n=1, it reduces to Bernoulli(p).
2. Normal Distribution Approximation: For large n, binomial(n,p) can be approximated by normal(μ=np, σ²=np(1-p)) when np > 5 and n(1-p) > 5. Use continuity correction for better accuracy.
3. Poisson Distribution Approximation: When n is large and p is small (but np is moderate, typically np < 10), binomial can be approximated by Poisson(λ=np).
4. Hypergeometric Distribution: Related but different; hypergeometric is for sampling without replacement from a finite population, while binomial assumes infinite population or replacement.
5. Negative Binomial Distribution: Counts trials until a fixed number of successes; binomial counts successes in fixed trials.
For n=100, p=0.4, calculate P(X ≤ 50) using normal approximation.
• μ = 100 × 0.4 = 40
• σ = √(100 × 0.4 × 0.6) = √24 ≈ 4.899
• Using continuity correction: P(X ≤ 50) ≈ P(X ≤ 50.5)
• Z = (50.5 - 40) / 4.899 ≈ 10.5 / 4.899 ≈ 2.143
• P(Z ≤ 2.143) ≈ 0.984
Exact binomial probability would be very close to this value.
Computational Methods and Tools
Calculating Binomial Probabilities
1. Direct Calculation: Use the PMF formula for small n. For larger n, this becomes computationally intensive due to large factorials.
2. Recursive Methods: Use Pascal’s triangle or recursive formulas to compute probabilities efficiently.
8 References
Academic Textbooks
- Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
- Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2016). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
- Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning.
- Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.
- Blitzstein, J. K., & Hwang, J. (2019). Introduction to Probability (2nd ed.). CRC Press.
- Sheldon, R. (2009). A First Course in Probability (8th ed.). Pearson Prentice Hall.
-
Khan Academy - Probability and Statistics:
Comprehensive free tutorials covering probability fundamentals, binomial distribution, and statistical inference.
https://www.khanacademy.org/math/statistics-probability -
MIT OpenCourseWare - Probabilistic Systems
Analysis:
Advanced probability theory course materials including lecture notes and problem sets.
https://ocw.mit.edu -
StatQuest with Josh Starmer:
YouTube channel with clear, intuitive explanations of probability and statistical concepts.
https://www.youtube.com/@statquest -
Seeing Theory - Brown University:
Interactive visualizations for probability and statistics concepts.
https://seeing-theory.brown.edu
Research Articles and Papers
- Kolmogorov, A. N. (1933). Foundations of the Theory of Probability. Chelsea Publishing Company.
- Bernoulli, J. (1713). Ars Conjectandi (The Art of Conjecturing). Basel: Thurneysen Brothers.
- Bayes, T., & Price, R. (1763). “An Essay towards solving a Problem in the Doctrine of Chances.” Philosophical Transactions of the Royal Society of London, 53, 370-418.
Documentation Note:
All visualizations,
examples, and interactive demonstrations in this document were created
specifically for educational purposes. The binomial distribution
formulas, probability concepts, and statistical methods are based on
standard probability theory as presented in the referenced textbooks and
academic literature. The graphical representations were generated using
modern data visualization tools to enhance understanding of theoretical
concepts.
This comprehensive study of probability essentials was prepared as part of the Week 10 assignment for the probability and statistics course. Special appreciation goes to:
- Course instructors for providing foundational knowledge and guidance throughout the semester
- Academic resources and online learning platforms that made advanced statistical concepts accessible
- The mathematical and statistical community for developing robust probability theory frameworks
- Open-source visualization tools that enabled clear presentation of complex probabilistic concepts