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Chapter 11: Problem 59

Find the probability. You roll a die. What is the probability that you will roll a four?

Short Answer

Expert verified
The probability of rolling a four on a six-sided die is 1/6.

Step by step solution

01

Identify the Total Outcomes

The die being used is a standard six-sided die. This means there are six total outcomes, as each side of the die has a unique number (1, 2, 3, 4, 5, 6). So, the total outcomes of rolling the die is 6.
02

Identify the Desired Outcome

We desire to roll a '4'. This is a single outcome and is only found on one face of the die out of six. So, the desired outcome is 1.
03

Calculate the Probability

Probability is the likelihood of a specific outcome or event happening. It is calculated as the ratio of the desired outcomes to the total outcomes. In this case, it is the ratio of rolling a '4' (desired outcome) to all possible outcomes of the die roll (total outcomes). Hence, probability = desired outcomes / total outcomes = 1 / 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dice
In the realm of probability, a dice roll is one of the most common examples used to illustrate basic principles. A standard die, often used in board games, is a cube with six faces. Each face is marked with a different number of dots ranging from 1 to 6.

Each face represents a different outcome, and because the die is fair, each number is equally likely to appear when you roll it. This concept of equally likely outcomes is important because it forms the basis for calculating probability. When working with dice, always acknowledge the number of sides and remember each side represents one potential outcome.
Total Outcomes
Understanding total outcomes is key to calculating probability. For a standard six-sided die, there are six possible outcomes since each side of the die shows a unique number from 1 to 6.

In probability, all these possible outcomes collectively represent what is known as the sample space. When you roll the die, any of the numbers 1 through 6 can show up. Therefore, the total number of outcomes is 6.

Knowing this allows us to set up our probability calculations, as we'll later compare the number of favorable outcomes to this total.
Desired Outcome
Identifying the desired outcome is essential when calculating probability. A desired outcome is the specific result you are interested in achieving.

For the exercise, our desired outcome is rolling a '4'. This is just one of the six possible outcomes on the die. The key part to note here is that the desired outcome is singular, meaning only rolling a number '4' will satisfy our condition.

Recognizing the simplicity or complexity of finding this in a roll helps determine how likely or unlikely it is to happen.
Calculating Probability
Probability is a measure of how likely an event is to occur. To calculate it, we use the formula: \[ \text{Probability} = \frac{\text{Desired Outcomes}}{\text{Total Outcomes}} \]In our dice example, the desired outcome is rolling a '4', and this occurs on just one face of the die.

Therefore, our calculation becomes:\[ \text{Probability of rolling a '4'} = \frac{1}{6} \]This fraction \( \frac{1}{6} \) represents a 16.67% chance of rolling a '4' in any single roll.

Calculating probability in this way gives us a numerical value that quantifies our understanding of the likelihood of specific outcomes.

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