Overview
This lecture covers the Probability chapter in a one-shot manner, including basic definitions, sample space, events, formulas, and tricks to solve exam-point questions.
Types of Probability
- Probability has two broad divisions: Subjective (experience-based) and Objective (based on mathematical rules).
- Objective probability is solved using mathematical rules.
Basic Terms and Sample Space
- Random Experiment: An experiment whose outcome cannot be determined beforehand (e.g., tossing a coin).
- Sample Space: The set of all possible outcomes.
- Coin: Sample space in n coin tosses = 2тБ┐.
- Dice: Sample space in n dice rolls = 6тБ┐.
- Playing Cards: Total 52; 26 red (hearts, diamonds), 26 black (spades, clubs); 12 face cards (K, Q, J).
Events and Their Categories
- Event = a subset of the sample space (e.g., getting an even number on dice).
- Simple (Elementary) Event: Cannot be broken down further.
- Compound/Composite Event: Can be divided into smaller events.
- Sure Event: Certain to occur (probability 1).
- Impossible Event: Cannot occur (probability 0).
Probability Formula and Tricks
- Probability = (Favorable outcomes) / (Total outcomes)
- Probability of an event not occurring = 1 - Probability of that event.
- Odds in favor: favorable/unfavorable; Odds against: unfavorable/favorable.
- "At least one": 1 - (probability of no favorable outcomes).
Important Event Relations
- Mutually Exclusive: Events with no common outcome.
- Exhaustive: Events whose union covers the entire sample space.
- Equally Likely: All have the same probability.
- Three mutually exclusive and exhaustive events: P(A)+P(B)+P(C)=1
Union and Intersection Formulas
- P(A тИк B) = P(A) + P(B) - P(A тИй B)
- P(A тИк B тИк C) = PA + PB + PC - (PAB + PBC + PCA) + PABC
Compound and Conditional Probability
- Compound Probability: Two or more events occurring together.
- Independent: P(A тИй B) = P(A) ├Ч P(B)
- Dependent: P(A тИй B) = P(A) ├Ч P(B|A)
- Conditional Probability: P(A|B) = P(A тИй B) / P(B)
Variance and Expectation
- Expectation: E(X) = ╬г x├ЧP(x)
- Variance: Var(X) = E(X┬▓) тАУ [E(X)]┬▓
Key Terms & Definitions
- Sample Space тАФ The set of all possible outcomes.
- Event тАФ A part of the sample space; a specific outcome.
- Mutually Exclusive тАФ Two events whose intersection is zero.
- Conditional Probability тАФ Probability of an event given another has occurred.
- Odds in Favor / Against тАФ Ratio of favorable to unfavorable outcomes.
- Expectation тАФ Expected average value.
- Variance тАФ Measure of spread of values.
Action Items / Next Steps
- Follow new targets from April 9.
- Join the 500 Most Important Question series from April 11.
- Homework: Practice the questions done in the lecture by yourself.
- In the upcoming session, study theoretical distribution.