Problem 33 - Algebra and Trigonometry

Question

The sample space is \(S=\\{1,2,3,4,5,6,\) 7,8,9,10}. Suppose that the outcomes are equally likely. Compute the probability of the event \(E:\) "an even number."

Step-by-Step Solution

Answer

The probability of selecting an even number is 0.5.

Step 1: Identify the Sample Space

The sample space, denoted as \(S\), includes all possible outcomes. Here, the sample space is given by: \(S=\{1,2,3,4,5,6,7,8,9,10\}\).

Step 2: Determine the Total Number of Outcomes

Count the total number of elements in the sample space \(S\). There are 10 elements in \(S\).

Step 3: Identify the Favorable Outcomes

List all outcomes that satisfy event \(E\), which is 'an even number'. The even numbers within the sample space \(S\) are \(\{2, 4, 6, 8, 10\}\).

Step 4: Compute the Number of Favorable Outcomes

Count the number of elements in the set of favorable outcomes. There are 5 even numbers, so there are 5 favorable outcomes.

Step 5: Calculate the Probability

The probability of event \(E\) is given by the ratio of the number of favorable outcomes to the total number of outcomes. Thus, the probability of selecting an even number is: \[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{10} = 0.5\]

Key Concepts

Sample Space

The sample space represents all possible outcomes in a probability experiment. For the given exercise, the sample space is: \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).
Each number in this set is a possible outcome when an experiment is conducted, such as randomly picking a number from 1 to 10.
Understanding the sample space is crucial as it forms the basis for determining probabilities.
Every probability problem starts with defining this set to clarify what outcomes are possible and what further calculations will be based on.

Favorable Outcomes

Favorable outcomes are the specific outcomes of the sample space that satisfy the event being investigated.
For the event \(E\) which describes 'an even number', we focus on those numbers in the sample space that are even.
In this case, the favorable outcomes in the sample space \(S\) are \(\{2, 4, 6, 8, 10\}\).
There are 5 favorable outcomes in total, since these are the numbers among 1 to 10 that meet the criteria of being even.
Identifying these outcomes is essential for the next steps in calculating the probability.

Equally Likely Outcomes

In many probability problems, including this one, it is assumed that all outcomes in the sample space are equally likely.
This means that each number in the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) has the same chance of being chosen.
When outcomes are equally likely, the probability of an event can be calculated effectively using the ratio of favorable outcomes to total outcomes.
This assumption simplifies calculations and is a common scenario in basic probability exercises.

Probability Calculation

To calculate the probability of an event, we use the formula:
\( P(E) = \frac{\text{Number of favorable outcomes}} {\text{Total number of outcomes}} \).
Here, for the event \(E\) which represents 'an even number', we determined there are 5 favorable outcomes out of a total of 10 possible outcomes.
Thus, the calculation is:
\( P(E) = \frac{5}{10} = 0.5\).
Therefore, the probability of selecting an even number from the sample space is 0.5 or 50%.
This step-by-step approach is crucial to clearly understand how probabilities are derived from specific conditions and sample spaces.