Problem 9 - College Algebra

Question

The sample space of equally likely outcomes is \(\\{1,2,3,4,5,6\\} .\) Find the probability of getting: a 4.

Step-by-Step Solution

Answer

The probability of getting a 4 is \( \frac{1}{6} \).

Step 1: Define the problem space

We know that a dice has 6 sides, with the numbers 1 to 6. So, when rolling the dice, we can get any one of these numbers. Each of these 6 outcomes is equally likely. Therefore, our sample space is \( \{1,2,3,4,5,6\} \), which means we have a total of 6 possible outcomes.

Step 2: Define the event

We are asked to find the probability of getting a 4. This is a single event, so it only consists of the number 4. This means that there is only 1 favorable outcome out of the 6 possible outcomes.

Step 3: Calculate the probability

Probability is calculated as the number of favorable outcomes divided by the total number of outcomes. So the probability of getting a 4 is \( \frac{1}{6} \).

Key Concepts

Sample Space

In probability, a sample space is a fundamental concept that represents all the possible outcomes of a random experiment. For instance, when rolling a standard six-sided dice, each face is numbered from 1 to 6. Thus, the sample space for this dice-rolling experiment is \( \{1, 2, 3, 4, 5, 6\} \). The sample space includes every potential result that could occur when the dice is rolled.

Understanding the concept of a sample space helps set the stage for calculating probabilities. It is essential because it defines the scope of the problem space.

• Each element, or event, in the sample space is a distinct outcome that can occur.
• For a dice, this means any number between 1 and 6 can appear on the top face.
• The completeness of the sample space ensures no result is overlooked in probability calculations.

Equally Likely Outcomes

Outcomes in a sample space are considered equally likely if each outcome has the same probability of occurring. This is a key assumption when working with fair random devices like a fair dice or a standard coin.

In the case of a six-sided dice, all outcomes \( \{1, 2, 3, 4, 5, 6\} \) are equally likely, as each side has an equal chance of landing face up. Therefore, the probability of rolling any single number on a fair dice is the same for each number.

• With a fair six-sided dice, each number has a probability of \( \frac{1}{6} \).
• Equally likely outcomes simplify the process of probability calculation since each outcome's chance of occurring is identical.
• This uniformity is crucial for making accurate predictions in probabilistic scenarios.

When it comes to real-life applications, assuming equally likely outcomes allows for straightforward probability computations. However, it's important to ensure the conditions reflect this assumption.

Favorable Outcome

The concept of a favorable outcome is central when determining probabilities. A favorable outcome is any one of the outcomes in the sample space that fulfills the criteria of the event we're interested in.

Consider, for example, rolling a dice and wanting to roll a 4. Only the outcome \(4\) satisfies this event, so there is just one favorable outcome here among the sample space's six possible outcomes.

• To compute the probability of a favorable outcome, count how many outcomes satisfy the event's condition.
• Divide this count by the total number of outcomes in the sample space.
• Using the earlier example, there's only 1 favorable result, and thus the probability is \( \frac{1}{6} \).

Comprehending favorable outcomes is essential for solving a wide array of probability problems correctly. Whenever you face a new probability question, define what your favorable outcomes will be before crunching the numbers.