Motivation for Probability
From games of chance to a rigorous mathematical theory
📜 Origins
Probability theory didn't start in a classroom. It started in casinos and gambling houses!
Gerolamo Cardano
Italian (1501–1576)
A gambling scholar who wrote Liber de Ludo Aleae (Book on Games of Chance), the first systematic treatment of probability.
Blaise Pascal
French (1623–1662)
Laid the foundation of modern probability theory through correspondence with Fermat about gambling problems.
Pierre-Simon Laplace
French (1749–1827)
Formalized classical probability:
\( P(E) = \frac{\text{Favorable Cases}}{\text{Total Cases}} \)
📖 Key Definitions
Random Experiment
A process with an uncertain outcome (e.g., tossing a coin, rolling a die).
Sample Space (\(S\))
The set of all possible outcomes.
Example: For a die roll, \(S = \{1, 2, 3, 4, 5, 6\}\)
Event (\(E\))
A subset of the sample space.
Example: Rolling an even number, \(E = \{2, 4, 6\}\)
⚖️ Axioms of Probability
// Kolmogorov's Axioms
- 1. For any event \(A\), \(P(A) \geq 0\)
- 2. \(P(S) = 1\) (Something must happen)
-
3.
If \(A_1, A_2, \dots\) are disjoint, then
\(P(\cup A_i) = \sum P(A_i)\)
Interactive Proof: Probability of Complement
Prove that \(P(A^c) = 1 - P(A)\).
📉 The Galton Board
The Galton Board demonstrates how the Binomial Distribution (and eventually the Normal Distribution) arises from simple random events.
At each peg, a ball has a 50/50 chance of going left or right. This is like a coin flip!
Connection to Pascal's Triangle:
The number of paths to reach a bin corresponds exactly to the numbers in Pascal's Triangle!
📝 Test Your Understanding
Question 1:
If \(P(A) = 0.3\) and \(P(B) = 0.4\), and \(A\) and \(B\) are disjoint (mutually exclusive), what is \(P(A \cup B)\)?
Question 2:
What is the probability of getting at least one Head in 3 coin tosses?