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To comprehend the idea of equally likely outcomes, imagine each possibility in an event having an identical chance of happening. For example, consider rolling a fair six-sided die. Each face is equally likely to land face up. Similarly, in our given problem, the outcomes \(E_1, E_2, E_3,\) and \(E_4\) are four events that are equally likely. This means each has an identical probability of occurring when the experiment is conducted.

In an experiment with equally likely outcomes, the total probability is always equal to 1. This is because the sum of the probabilities of all potential independent possibilities must equal certainty—one complete event must occur. If you sum all these equal probabilities, you'll reach a total of 1:
\[P(E_1) + P(E_2) + P(E_3) + P(E_4) = 1\].
Each outcome then has an individual probability of \(\frac{1}{4}\) in our scenario, reflecting the fair and balanced chance of each event happening.
Understanding these principles aids in calculating probabilities, especially in scenarios with equally likely outcomes.

Calculating probability is about determining how likely an event is to occur. Fundamentally, probability measures the ratio of favorable outcomes to the total number of possible outcomes.

To calculate the probability of a single event, such as \(E_2\), in an experiment where each outcome is equally probable, you divide the number of ways \(E_2\) can occur (which is 1 event) by the total number of outcomes (which is 4 outcomes). The probability is:
\[P(E_2) = \frac{1}{4}\].

When calculating the probability of multiple events like \(E_1\) or \(E_3\), you'll add their individual probabilities. Since each individual event has a probability of \(\frac{1}{4}\), the combined probability of any two events occurring is:
\[P(E_1 \text{ or } E_3) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}\].
Consider the probability of three events happening such as \(E_1, E_2, \text{ or } E_4\). By adding their probabilities, you achieve:
\[P(E_1 \text{ or } E_2 \text{ or } E_4) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}\].
This straightforward addition arises from the fact that the outcomes are equally probable.
This simplicity is the beauty of equally likely outcomes, making probability calculation a straightforward process.

In probability, an event refers to any outcome or a group of outcomes from an experiment. Understanding event outcomes is essential as it forms the base of probability theory which deals with evaluating how likely it is for an event to happen.

Outcomes like \(E_1, E_2, E_3,\) and \(E_4\) from our experiment can be combined to form more complex events. For example, selecting any one, two, or three of these outcomes forms different events. Logical thinking about outcomes involves ensuring each scenario is accounted for and correctly interpreted.
Let's discuss various outcome combinations:

• A single outcome event: If only \(E_2\) is considered, we simply find its probability as a single occurrence.
• A two-outcome event: Combining outcomes like \(E_1\) or \(E_3\) creates a multi-outcome event probability.
• A three-outcome event: More complex events like \(E_1, E_2, \text{ or } E_4\) cover a majority of the possibilities, increasing the likelihood.

These combinations demonstrate that events can range from simple occurrences to comprehensive gatherings of multiple possibilities. Recognizing this range helps in accurately predicting and calculating probabilities in various scenarios.