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The concept of a sample space is fundamental in probability theory. It represents the complete set of all possible outcomes of a particular experiment. If you think about conducting an experiment, the sample space composes all the results you might get.
In our given exercise, the sample space is denoted by the set \( S=\{s_1, s_2, s_3, s_4, s_5\} \). This means there are five possible outcomes when the experiment is conducted, and they are labeled as \( s_1, s_2, s_3, s_4, \) and \( s_5 \).
This is crucial because it helps set the boundaries for what the probability distribution will need to encompass.

A probability distribution assigns a probability to each outcome in the sample space. This tells us how likely each result is when an experiment is carried out. The sum of all assigned probabilities should equal 1, demonstrating that they account for the totality of outcomes.
For instance, in the exercise provided, the probability distribution is shown across the outcomes \( s_1 \) through \( s_5 \):

• \( P(s_1) = \frac{1}{14} \)
• \( P(s_2) = \frac{3}{14} \)
• \( P(s_3) = \frac{6}{14} \)
• \( P(s_4) = \frac{2}{14} \)
• \( P(s_5) = \frac{2}{14} \)

Each of these probabilities represent the relative likelihood of their respective outcomes occurring. Summing them, we have, \( \frac{1}{14} + \frac{3}{14} + \frac{6}{14} + \frac{2}{14} + \frac{2}{14} = 1 \).
This confirms the validity of the probability distribution as a model for the sample space.

Event probability refers to the likelihood of a specific set of outcomes occurring. To find it, we simply add up the probabilities of all the individual outcomes that make up the event.
When solving for event probability, like with events \( A \), \( B \), and \( C \) from the exercise, you identify which outcomes form the event then sum their probabilities.
For event \( A \), consisting of outcomes \( \{s_1, s_2, s_4\} \), the probability is:

• \( P(A) = P(s_1) + P(s_2) + P(s_4) = \frac{1}{14} + \frac{3}{14} + \frac{2}{14} = \frac{6}{14} \)

By doing this, you calculate how likely it is for the particular event (involving these specific outcomes) to occur when the experiment is performed.
This process is repeated similarly for events \( B \) and \( C \).

Outcomes in probability are the individual elements or results within a sample space after an experiment is conducted. Recognizing and understanding the implications of these outcomes help us build towards calculating probabilities.
In our exercise, each outcome is different (\( s_1, s_2, s_3, s_4, s_5 \)), and each can be part of different events.
If you imagine rolling a fair die, each face coming up (from 1 to 6) would be considered an outcome.
To solve cases involving probability, determine each possible outcome first. Then, you can start exploring and calculating associated probabilities by employing the defined sample space and its probability distribution.