Problem 10 - Basic College Mathematics

Question

Find each of the following probabilities. Rolling a Die. In Exercises \(7-12,\) assume that one die is rolled. Find the probability that a number greater than 2 is rolled.

Step-by-Step Solution

Answer

The probability of rolling a number greater than 2 is \(\frac{2}{3}\).

Step 1: Understand the Die Faces

A standard die has 6 faces, numbered from 1 to 6. Each face is equally likely to appear when the die is rolled.

Step 2: Identify the Favorable Outcomes

The problem asks for the probability of rolling a number greater than 2. The numbers greater than 2 on a die are 3, 4, 5, and 6. Thus, there are 4 favorable outcomes.

Step 3: Calculate the Total Possible Outcomes

Since the die has 6 faces, there are 6 possible outcomes in total when rolling the die.

Step 4: Use the Probability Formula

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. So, we calculate the probability using the formula: \[ P(\text{number > 2}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{4}{6} \ = \frac{2}{3} \ \therefore P(\text{number > 2}) = \frac{2}{3} \]

Key Concepts

rolling a die

Rolling a die refers to the simple act of throwing a six-sided die, a common cube-shaped object used in many games. Each of the six faces of the die has unique numbers ranging from 1 to 6. When the die is rolled, it lands with one face up, showing a random number. Because each face is equally likely to land up, the outcome of rolling a die is inherently random.
Understanding the concept of rolling a die is the base for grasping probability problems related to dice. Each time the die is rolled, there is no bias towards any particular number, ensuring each face has an equal chance.

favorable outcomes

In probability, a 'favorable outcome' means an outcome that satisfies the condition of a particular event. For example, if we are interested in finding the probability of rolling a number greater than 2 on a die, we identify which numbers on the die are greater than 2.
For a standard six-sided die, the numbers greater than 2 are 3, 4, 5, and 6. There are four favorable outcomes here because these are the outcomes that meet our condition.
In general, determining favorable outcomes means picking out the results that align with the event we are interested in predicting.

probability formula

The probability of an event happening can be calculated using a straightforward formula. This formula compares the number of favorable outcomes to the total number of possible outcomes. It is written as:
\[ P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
Using our previous example:

• We have 4 favorable outcomes (rolling a 3, 4, 5, or 6).
• The total number of possible outcomes, since a die has 6 faces, is 6.

So, the probability of rolling a number greater than 2 is calculated as:\[ P(\text{number > 2}) = \frac{4}{6} = \frac{2}{3} \]
By using this formula, we can systematically and accurately determine the likelihood of different events when rolling a die or in other probability scenarios.