Problem 11 - College Algebra with Modeling and Visualization

Question

Find the probability of each event. Rolling a 2 with a fair die

Step-by-Step Solution

Answer

The probability of rolling a 2 is \( \frac{1}{6} \).

Step 1: Understanding the Setup

We are rolling a fair 6-sided die. Each side of the die is equally likely to come up on any roll. We need to calculate the probability of rolling a 2.

Step 2: Determine Total Outcomes

A fair die has six faces. Therefore, there are 6 possible outcomes when you roll the die: 1, 2, 3, 4, 5, and 6.

Step 3: Identify Favorable Outcome

The favorable outcome is rolling a 2. There is only 1 face with the number 2 on the die.

Step 4: Apply the Probability Formula

The probability of an event is calculated using the formula: \[ P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \] For this example, the number of favorable outcomes is 1 (rolling a 2) and the total number of possible outcomes is 6 (all die faces).

Step 5: Calculate the Probability

Substitute the numbers into the probability formula: \[ P(\text{rolling a 2}) = \frac{1}{6} \]

Key Concepts

Probability Formula

Understanding the probability formula is key to solving problems involving the likelihood of events. Probability measures how likely an event is to happen compared to all possible events. To find this, we apply the probability formula, which is:\[ P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]In this formula:

• P(E) represents the probability of an event E occurring.
• The Number of Favorable Outcomes is how many ways the event of interest can happen.
• The Total Number of Possible Outcomes is all possible events that can occur in the setup.

This formula gives a simple fractional representation of probability that can be converted to a decimal or percentage if needed.

Probability Calculation

Calculating probability involves using the formula with specific numbers from a given scenario. First, identify the favorable outcomes, which are the outcomes of interest. Then, find out the total possible outcomes. For instance, when rolling a fair die:- The favorable outcome of rolling a 2 is 1, since there's one face showing a 2.- The total possible outcomes refer to all six faces of the die.Using the formula, the probability calculation for rolling a 2 would be:\[ P(\text{rolling a 2}) = \frac{1}{6} \]This calculation shows there’s a 1 out of 6 chance, or approximately 16.67%, of rolling a 2.

Understanding Outcomes

Understanding outcomes is critical in probability as it defines the foundation of any probability-related problem. In any scenario, we categorize outcomes into two parts:

• Favorable Outcomes: These are the specific outcomes we are interested in. For a die roll when you're hoping for a 2, the favorable outcome is rolling a 2 itself, just one scenario among many.
• Possible Outcomes: These encompass all the outcomes that might happen when uncertain events are considered. For a die, it can land on any one of its six faces, making six possible outcomes.

Recognizing and organizing outcomes helps simplify the problem-solving process by clearly defining what is being calculated. Proper understanding of outcomes ensures correct application of the probability formula, helping avoid common errors.