This topic covers the basic concepts of probability, including sample spaces, events, and probability calculations for simple and compound events.
Introduction to Probability – Study Notes
| Concept | Description |
|---|---|
| Probability | A numerical measure of how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). |
| Experiment | Any process that generates outcomes (e.g., rolling a die, drawing a card). |
| Sample Space ( S ) | The set of all possible outcomes of an experiment. |
| Event | A subset of the sample space (one or more outcomes). |
Example:
For a finite, equally likely sample space:
[ P(A) = \frac{\text{Number of outcomes in }A}{\text{Number of outcomes in }S} ]
Example: Probability of drawing an Ace from a standard 52‑card deck
[ P(\text{Ace}) = \frac{4}{52} = \frac{1}{13} ]
From these axioms follow useful rules:
| Type | Definition | Probability Rule |
|---|---|---|
| Simple | Single outcome (e.g., rolling a 3). | Direct count or formula. |
| Compound | Combination of outcomes (e.g., “roll an even number”). | Use addition rule. |
| Mutually exclusive | Cannot occur together (e.g., rolling a 2 and a 5). | (P(A\cup B)=P(A)+P(B)). |
| Independent | Occurrence of one does not affect the other. | (P(A\cap B)=P(A)P(B)). |
| Dependent | One event influences the other. | Use conditional probability. |
Probability of (A) given that (B) has occurred:
[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(B) > 0 ]
Events (A) and (B) are independent iff:
[ P(A \mid B) = P(A) \quad \text{or equivalently} \quad P(A \cap B) = P(A)P(B) ]
| Technique | When to Use | Formula |
|---|---|---|
| Multiplication Rule | Ordered selections, each step independent | (n_1 \times n_2 \times \dots \times n_k) |
| ** |