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:
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Define the Sample Space (S): Identify all the possible outcomes of the experiment. This set is known as the sample space.
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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.
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Count Favorable Outcomes: Count how many outcomes in the sample space are favorable to the event \( E \).
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Count Total Outcomes: Count the total number of outcomes in the sample space.
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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}).
- Number of favorable outcomes for E: 3 (the outcomes 2, 4, and 6).
- 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.