Sample Space
In probability theory, the sample space is a fundamental concept. It represents the set of all possible outcomes of a particular experiment. Let's consider the exercise given. Here, the sample space is defined as S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. This means that if we were to randomly select one number from this set, any of the numbers from 1 to 10 could be the result.
Each number in this set is an outcome. Because no outcomes were favored or neglected, each outcome is equally likely. This idea of sample space gives a complete picture of all possible outcomes for an event.
Even Numbers
Even numbers are integers that are exactly divisible by 2. In other words, when you divide an even number by 2, there is no remainder.
Let's identify the even numbers within our sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Looking through this set, we find that the even numbers are 2, 4, 6, 8, and 10.
These are the numbers that satisfy the condition of being divisible by 2 without any remainder. Therefore, out of the 10 numbers in the sample space, 5 of them are even numbers.
Favorable Outcomes
Favorable outcomes are those outcomes in the sample space that satisfy the condition of the event in question. In our exercise, the event E is defined as 'an even number'. This means that the favorable outcomes are those outcomes within our sample space that are even numbers.
From the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, the favorable outcomes are {2, 4, 6, 8, 10}.
There are 5 favorable outcomes in this case. Identifying these favorable outcomes is an important step in calculating the probability of an event.
Probability Formula
To calculate the probability of an event, we use a specific formula. The probability formula is given by:
\( P(E) = \frac{{\text{{Number of favorable outcomes}}}}{{\text{{Total number of outcomes}}}} \)
In our exercise, the event E is 'an even number'. We have already identified that there are 5 favorable outcomes and a total of 10 possible outcomes in the sample space. By substituting these values into the formula, we get:
\[ P(E) = \frac{5}{10} = 0.5 \]
This results shows that the probability of selecting an even number from the sample space S is 0.5, or 50%. The probability formula makes it simple to compute the likelihood of any event, given the necessary information.
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