Problem 3 - Algebra 2

Question

A die is rolled. Find each probability. \(P(\text { less than } 3)\)

Step-by-Step Solution

Answer

\(\frac{1}{3}\)

Step 1: Understand the Problem

A die has six faces, numbered from 1 to 6. We need to find the probability of rolling a number less than 3.

Step 2: Identify Favorable Outcomes

The numbers on a die that are less than 3 are 1 and 2. Therefore, there are 2 favorable outcomes.

Step 3: Identify Total Outcomes

Since a die has 6 faces, there are 6 possible outcomes when it is rolled.

Step 4: Calculate Probability

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. So, the probability of rolling a number less than 3 is \( \frac{2}{6} = \frac{1}{3} \).

Key Concepts

Favorable Outcomes

When trying to calculate the probability of an event, it's important to first determine the favorable outcomes. These are the outcomes that satisfy the condition we're interested in. In the context of rolling a die and looking for numbers less than 3, the favorable outcomes are the numbers that meet this condition.

For a standard die, the numbers are simply the integers from 1 to 6. When we look for numbers less than 3, we find the numbers 1 and 2 meet this requirement.

• Number 1
• Number 2

Thus, there are two favorable outcomes in this scenario. By identifying these outcomes, you set the stage for calculating the probability of the event.

Total Outcomes

In probability, the total outcomes refer to all possible results that can occur from the random event. For a single die thrown, these possibilities are very straightforward. In this case, since a standard die has six faces—each marked with a unique number from 1 to 6—the total outcomes when rolling the die are six.

• 1
• 2
• 3
• 4
• 5
• 6

Understanding and listing the total outcomes helps create the framework you need to compare to the favorable outcomes.

Event Probability

The probability of a particular event is a fundamental concept in probability theory, and it is determined by comparing favorable outcomes to total outcomes. For any event, the probability is calculated using the formula:\[ \text{Probability of an event} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]In our die-rolling example, we have already identified there are 2 favorable outcomes (1 and 2) and 6 total possible outcomes. Using the formula, the probability of rolling a number less than 3 is calculated as follows: \[ \frac{2}{6} = \frac{1}{3} \]This means there is a one-third chance, or about 33.33% probability, that a roll of the die results in a number less than 3. Remember, understanding how to break down a problem using these three steps makes calculating probabilities much easier.