Introduction to Sample Spaces and Events in Probability

Let's revisit the example of rolling a single six-sided die where $S = {1, 2, 3, 4, 5, 6}$. We can define several different events:

1. Event A (Rolling an even number): $A = {2, 4, 6}$
2. Event B (Rolling a number greater than 4): $B = {5, 6}$
3. Event C (Rolling a 7): $C = \emptyset$ (This is an "impossible event," represented by the empty set).

Because an event is a subset, it is mathematically represented using set notation. If every outcome in the sample space is included in an event, we call it a "certain event."

Relationships Between Events

In more advanced probability, we often look at how two or more events interact with one another. Understanding these relationships is crucial for calculating the probability of complex scenarios.

Intersection and Union

• Intersection ($A \cap B$): This represents the event where both Event A and Event B occur. In our die example, if $A = {2, 4, 6}$ and $B = {5, 6}$, then $A \cap B = \{6\}$.
• Union ($A \cup B$): This represents the event where either Event A occurs, Event B occurs, or both occur. Using the same example, $A \cup B = \{2, 4, 5, 6\}$.

Complementary Events

The complement of an event $A$, denoted as $A^c$ or $A'$, consists of all the outcomes in the sample space that are not in $A$.

If our event $A$ is "rolling an even number" ${2, 4, 6}$, then the complement $A^c$ is "rolling an odd number" ${1, 3, 5}$. A fundamental rule in probability is that the probability of an event plus the probability of its complement must always equal 1:

$$P(A) + P(A^c) = 1$$

Worked Example: The Two-Coin Toss

To solidify these concepts, let's walk through a complete problem.

Scenario: You toss two fair coins.

Step 1: Define the Sample Space ($S$) The possible outcomes are:

$$S = \{HH, HT, TH, TT\}$$

The total number of outcomes $n(S) = 4$.

Step 2: Define the Events

• Let Event E be "getting exactly one Head." Looking at our sample space, the outcomes with exactly one Head are $HT$ and $TH$. $E = {HT, TH}$
• Let Event F be "getting at least one Head." The outcomes with one or more Heads are $HH, HT,$ and $TH$. $F = {HH, HT, TH}$

Step 3: Analyze the Relationship What is the intersection of $E$ and $F$?

$$E \cap F = \{HT, TH\}$$

In this case, $E$ is actually a subset of $F$ ($E \subset F$), because every outcome that satisfies "exactly one head" also satisfies "at least one head."

Summary Checklist

• Did I list every possible outcome for the Sample Space?
• Is my Event a valid subset of that Sample Space?
• Am I distinguishing between "and" (intersection) and "or" (union)?
• Have I checked if my event is impossible ($\emptyset$) or certain ($S$)?