Question

The probability of an event occuring based on mathematical reasoning and the assumption that all outcomes are equally likely.

Answer 1

The probability of an event occurring can be calculated using a fundamental principle in probability theory, especially under the assumption that all outcomes are equally likely. In this context, the probability \( P \) of an event \( E \) occurring is defined as:

\[
P(E) = \frac{\text{Number of favorable outcomes for } E}{\text{Total number of possible outcomes}}
\]

### Steps to Calculate Probability:

1. **Define the Sample Space (S)**: Identify all the possible outcomes of the experiment. This set is known as the sample space.

2. **Identify the Event (E)**: Determine the specific event for which you want to calculate the probability, which consists of one or more outcomes from the sample space.

3. **Count Favorable Outcomes**: Count how many outcomes in the sample space are favorable to the event \( E \).

4. **Count Total Outcomes**: Count the total number of outcomes in the sample space.

5. **Calculate Probability**: Use the formula above to find the probability.

### Example:

Suppose you roll a fair six-sided die.

- **Sample Space (S)**: {1, 2, 3, 4, 5, 6} (total of 6 outcomes)
- **Event (E)**: Rolling an even number (which includes {2, 4, 6}).

1. **Number of favorable outcomes for E**: 3 (the outcomes 2, 4, and 6).
2. **Total number of possible outcomes**: 6.

Using the probability formula:

\[
P(E) = \frac{3}{6} = \frac{1}{2}
\]

Thus, the probability of rolling an even number on a six-sided die is \( \frac{1}{2} \).

This method applies broadly to many different situations where outcomes can be defined and counted.