1. The Language of Probability
Core Vocabulary
| Term | Definition | Example |
|---|---|---|
| Random Experiment | A repeatable process whose outcome cannot be predicted with certainty. | Rolling a die, flipping a coin, drawing a card. |
| Elementary Outcome | A single, indivisible result of the experiment (also called a sample point). | Rolling a 4; drawing the Queen of Hearts. |
| Sample Space ($S$) | The set of all elementary outcomes. | $S_{\text{die}} = \{1,2,3,4,5,6\}$ |
| Event | Any subset of the sample space. | "Rolling an even number" $= \{2,4,6\}$ |
🔑 Golden Rule: Always write out the sample space before any computation. This single habit prevents the majority of probability errors.
Concrete Sample Spaces
One Fair Die
$$S = \{1, 2, 3, 4, 5, 6\}$$
$|S| = 6$ elementary outcomes, each equally likely.
Two Coins
$$S = \{HH, HT, TH, TT\}$$
$|S| = 4$. Note: $HT \neq TH$ — order matters because the coins are distinguishable (first coin, second coin).
One Card from a Standard Deck
$$|S| = 52$$
Each card is a distinct elementary outcome: Ace of Spades, 2 of Spades, …, King of Diamonds.
2. The Algebra of Events
Events are sets, so we manipulate them with set operations. Each operation has a plain-English translation.
| Operation | Notation | Meaning | English |
|---|---|---|---|
| Union | $A \cup B$ | All outcomes in $A$ or $B$ (or both) | "At least one of $A$, $B$ occurs" |
| Intersection | $A \cap B$ | Outcomes in both $A$ and $B$ | "Both $A$ and $B$ occur" |
| Complement | $A'$ (or $\overline{A}$) | All outcomes NOT in $A$ | "$A$ does not occur" |
| Certain Event | $S$ | The entire sample space | "Something happens" |
| Impossible Event | $\emptyset$ | The empty set | "Nothing happens" (contradiction) |