Understanding Events and How They Affect Probability

What is an Event?
In probability, an event is an outcome or a group of outcomes from a random experiment.

For example, if you roll a six-sided die, some possible events are:

• Rolling a 3
• Rolling an even number (like 2, 4, or 6)

What is Sample Space?
The sample space, shown as $S$, includes all possible outcomes of an experiment.

For the die, the sample space looks like this:

$$S = \{1, 2, 3, 4, 5, 6\}$$

This tells us that there are 6 possible outcomes when rolling the die.

Types of Events
There are two main types of events:

1. Simple Event: This is just one outcome. For example, rolling a 4.

2. Compound Event: This includes two or more outcomes. For example, rolling an even number.

How to Calculate Probability
To find the probability of an event $E$, you can use this formula:

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

Using our die example, the probability of rolling an even number is:

$$P(\text{Even}) = \frac{3}{6} = \frac{1}{2} \approx 0.5$$

How Events Influence Probability
Events help us understand probability by showing how likely something is to happen. Generally, if there are more favorable outcomes for an event, its probability goes up.

For example, if you have a bag with 3 red and 2 blue marbles, the chance of drawing a red marble is:

$$P(\text{Red}) = \frac{3}{5} = 0.6$$

So, events not only help us figure out what can happen, but they also show how likely those outcomes are.