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Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means the event will certainly happen. In our experiment, since each outcome is equally likely, this makes calculating the probability straightforward. For any event, the probability can be calculated using the formula
\[ P(E) = \frac{|E|}{|S|} \] where

P(E)

represents the probability of event E,

|E|

represents the number of favorable outcomes in event E, and

|S|

represents the total number of outcomes in the sample space S. So, to find the probability of our events, we only need to count the number of outcomes in those events and divide by the total number of outcomes in the sample space, S.

The sample space (S) is the set of all possible outcomes in a probability experiment. It acts as a foundation on which probabilities are calculated. In our given problem, the sample space is:
\[ S=\{1,2,3,4,5,6,7,8,9,10,11,12\} \] Each of these numbers represents an outcome of the experiment. Let's look at events inside this sample space:

Event E = {2, 3, 4, 5, 6, 7}

This includes outcomes between 2 and 7.

Event F = {5, 6, 7, 8, 9}

This includes outcomes between 5 and 9. These events are subsets of the sample space. Understanding the sample space helps us see that events are just subsets of it, and their probability is based on their size relative to the size of the sample space.

Two events are mutually exclusive if they cannot happen at the same time. This means they have no common outcomes. To determine mutual exclusivity, we check for shared elements between events.
In our exercise:

Event F = {5, 6, 7, 8, 9}

Event H = {2, 3, 4}

There are no shared elements. Thus, F and H are mutually exclusive events.
Understanding mutual exclusivity is crucial in probability because if two events are mutually exclusive, their combined probability is just the sum of their individual probabilities. For example:
P(F or H) = P(F) + P(H)

When outcomes are equally likely, each outcome has the same chance of occurring. For example, in our sample space S, every number from 1 to 12 is equally likely to be selected.
This simplifies calculating probabilities because each outcome has a probability of

\( \frac{1}{|S|} \)

.
\( \frac{1}{12} \)
This uniformity is essential for many probability calculations because it allows us to directly count how many outcomes fit an event and relate this count to the total number.
To recap:
Equally likely outcomes condition means probability calculations mainly involve counting favorable outcomes and dividing by the total possible outcomes. This canonical approach plays a fundamental role in basic probability theory.