Finding Classical Probabilities In Exercises 41-46, a probability...

4. Probability

Basic Concepts of Probability

Statistics

4. Probability

Basic Concepts of Probability

2:11 minutes

Problem 3.1.46

Textbook Question

Finding Classical Probabilities In Exercises 41-46, a probability experiment consists of rolling a 12-sided die numbered 1 to 12. Find the probability of the event.
46. Event F: rolling a number divisible by 5

Verified step by step guidance

Step 1: Understand the problem. The experiment involves rolling a 12-sided die numbered from 1 to 12. The goal is to find the probability of rolling a number divisible by 5.

Step 2: Identify the numbers divisible by 5 within the range of 1 to 12. A number is divisible by 5 if it can be expressed as 5 × k, where k is an integer. In this case, the numbers divisible by 5 are 5 and 10.

Step 3: Count the total number of outcomes in the sample space. Since the die has 12 sides, the total number of outcomes is 12.

Step 4: Count the favorable outcomes for Event F. The favorable outcomes are the numbers divisible by 5, which are 5 and 10. Therefore, there are 2 favorable outcomes.

Step 5: Calculate the probability using the classical probability formula: \( P(F) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Substitute the values: \( P(F) = \frac{2}{12} \). Simplify the fraction if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Classical Probability

Classical probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in a probability experiment. It assumes that all outcomes are equally likely. For example, when rolling a 12-sided die, the total outcomes are 12, and the probability of rolling a specific number is 1/12.

Divisibility

Divisibility is a mathematical concept that determines whether one integer can be divided by another without leaving a remainder. In the context of the given problem, we are interested in the numbers on the die that are divisible by 5, which are 5 and 10. Understanding which numbers meet this criterion is essential for calculating the probability of event F.

Event and Sample Space

An event is a specific outcome or a set of outcomes from a probability experiment, while the sample space is the set of all possible outcomes. In this case, the sample space consists of the numbers 1 through 12 on the die, and event F includes the outcomes that are divisible by 5. Identifying the event and sample space is crucial for calculating the probability of event F.

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Finding Classical Probabilities In Exercises 41-46, a probability experiment consists of rolling a 12-sided die numbered 1 to 12. Find the probability of the event.46. Event F: rolling a number divisible by 5