Problem 33 - Algebra 2

Question

A die is rolled, Find each probability. \(P(1 \text { or } 6)\)

Step-by-Step Solution

Answer

The probability of rolling a 1 or 6 is \(\frac{1}{3}\).

Step 1: Understanding the Problem

We need to find the probability of rolling a 1 or a 6 with a die. A standard die has 6 sides, numbered from 1 to 6.

Step 2: Identifying Favorable Outcomes

The favorable outcomes are rolling a 1 or rolling a 6. Therefore, the favorable outcomes are 2 (rolling a 1 and rolling a 6).

Step 3: Determining Total Possible Outcomes

Since a die has 6 sides, the total possible outcomes are 6.

Step 4: Calculating the Probability

The probability of an event is calculated using the formula \(P(A) = \frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}\). Here, \(P(1 \text{ or } 6) = \frac{2}{6} = \frac{1}{3}\).

Step 5: Simplifying the Fraction

The fraction \(\frac{2}{6}\) can be simplified to \(\frac{1}{3}\) by dividing the numerator and the denominator by 2. So the probability \(P(1 \text{ or } 6) = \frac{1}{3}\).

Key Concepts

Favorable Outcomes

Let's dig into what we mean by favorable outcomes when discussing probability. In any given probability scenario, favorable outcomes are the outcomes of interest that we specifically want to occur. These are not just any random outcomes; they are events that satisfy the condition we are considering.
For example, in the scenario of rolling a die and wanting to get either a 1 or a 6, the favorable outcomes are exactly those rolls that result in a 1 or a 6.

• If you roll a 1, it counts as 1 favorable outcome.
• If you roll a 6, it counts as another favorable outcome.

Thus, for this exercise, we have 2 favorable outcomes: rolling a 1 or a 6.

Total Possible Outcomes

Total possible outcomes refer to the complete set of outcomes that could happen in an event. This set includes all conceivable results, regardless of whether they are favorable or not.
When you roll a standard six-sided die, each side represents a distinct outcome. So, you have:

• 1,
• 2,
• 3,
• 4,
• 5,
• 6.

In this situation, the total possible outcomes are all 6 sides of the die. This means that the die can land on any of its six faces, making the total possible outcomes equal to 6. Each outcome is equally likely when rolling a fair die.

Probability Calculation

Finally, let’s piece together everything by calculating the probability. Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event).
The formula for calculating the probability of a specific event is: \[P(A) = \frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}\]In our die-rolling example, we want the probability that the outcome is either a 1 or a 6.
1. Number of Favorable Outcomes: 22. Total Possible Outcomes: 6
Substituting these into our formula gives us:\[P(1 \text{ or } 6) = \frac{2}{6}\]Simplifying the fraction by dividing both the numerator and denominator by 2, we get:\[P(1 \text{ or } 6) = \frac{1}{3}\]This fraction can also be interpreted as a 33.3% probability, meaning you have about a one in three chance to roll a 1 or 6.