Understanding Probability: Events and Sample Spaces

In probability, we look at different outcomes using something called a sample space.

The sample space, shown as $S$, includes all possible results from a random experiment.

For example, if you roll a six-sided die, the sample space would be $S = \{1, 2, 3, 4, 5, 6\}$.

What Are Events?

An event is simply a part of the sample space.

For example, if you want to talk about rolling an even number, you could define that event as $E = \{2, 4, 6\}$.

How to Calculate Probability

To find the probability of an event $E$, we can use this formula:

$P(E) = \frac{|E|}{|S|}$

Here, $|E|$ means the number of outcomes that fit your event, and $|S|$ is the total number of outcomes in the sample space.

Key Rules of Probability

1. Addition Rule: If two events, $A$ and $B$, cannot happen at the same time (we call them mutually exclusive), then:

   $P(A \cup B) = P(A) + P(B)$

   This means you can just add their probabilities together.

2. Multiplication Rule: If two events, $A$ and $B$, can happen at the same time (we call them independent), then:

   $P(A \cap B) = P(A) \cdot P(B)$

   This means you multiply their probabilities together.

By understanding these concepts, you can start to grasp how probability works in everyday situations!