Probability Of Rolling A Multiple Of 3 Or A Total Of 10 A Comprehensive Guide

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Understanding probability is crucial in various fields, from mathematics and statistics to everyday decision-making. This article delves into a specific probability problem: What is the probability of getting either a multiple of 3 on the first roll or a total of 10 for both rolls when a die is rolled twice? This seemingly simple question involves understanding basic probability principles, identifying favorable outcomes, and applying the addition rule of probability. Whether you're a student learning probability for the first time or someone looking to brush up on your skills, this comprehensive guide will walk you through the solution step-by-step.

Before we tackle the main problem, let's establish a solid foundation by revisiting some fundamental concepts of probability.

Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The formula for calculating probability is straightforward:

Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

In the context of rolling a fair six-sided die, there are six possible outcomes: 1, 2, 3, 4, 5, and 6. Each outcome has an equal chance of occurring. Therefore, the probability of rolling any specific number is 1/6. Understanding this basic principle is essential for solving more complex probability problems.

Independent Events are events whose outcomes do not affect each other. For example, the result of the first roll of a die does not influence the result of the second roll. Each roll is an independent event. When dealing with independent events, we often need to consider the combined outcomes of multiple events, which brings us to the concept of sample space.

Sample Space is the set of all possible outcomes of an experiment. When rolling a die twice, the sample space consists of all possible pairs of numbers that can result from the two rolls. To visualize this, we can create a table or a grid. The table will have 6 rows and 6 columns, representing the outcomes of the first and second rolls, respectively. Each cell in the table represents a unique outcome, such as (1, 1), (1, 2), (2, 1), and so on. There are a total of 36 possible outcomes (6 outcomes for the first roll multiplied by 6 outcomes for the second roll), making the sample space crucial for calculating probabilities involving multiple events.

To solve our problem, we need to identify the favorable outcomes – those that satisfy the given conditions. The problem asks for the probability of getting either a multiple of 3 on the first roll or a total of 10 for both rolls. Let's break this down into two separate events:

  1. Event A: Rolling a multiple of 3 on the first roll.
  2. Event B: Rolling a total of 10 for both rolls.

Our goal is to find the probability of either event A or event B occurring. We'll start by listing the outcomes that satisfy each event.

Event A: Multiples of 3 on the First Roll

A multiple of 3 on a six-sided die can be either 3 or 6. So, for the first roll, we have two favorable outcomes. Now, we need to consider all possible outcomes for the second roll. Since the second roll can be any number from 1 to 6, the favorable outcomes for Event A are:

  • (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
  • (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

There are 12 outcomes in total that satisfy Event A. This is because there are 2 favorable outcomes for the first roll and 6 possible outcomes for the second roll (2 * 6 = 12).

Event B: Total of 10 for Both Rolls

Next, we identify the outcomes where the sum of the numbers on both rolls is 10. These outcomes are:

  • (4, 6)
  • (5, 5)
  • (6, 4)

There are 3 outcomes that satisfy Event B.

Overlapping Outcomes

Now, we need to consider if there are any outcomes that satisfy both events. This is crucial because we don't want to double-count these outcomes when calculating the overall probability. Looking at the outcomes for Event A and Event B, we see that the outcomes (6, 4) is present in both lists. This means that rolling a 6 on the first roll and a 4 on the second roll satisfies both conditions: it's a multiple of 3 on the first roll, and the total of both rolls is 10.

To find the probability of either Event A or Event B occurring, we use the Addition Rule of Probability. This rule states that the probability of either of two events occurring is the sum of their individual probabilities, minus the probability of both events occurring. The formula is:

P(A or B) = P(A) + P(B) - P(A and B)

Where:

  • P(A or B) is the probability of either event A or event B occurring.
  • P(A) is the probability of event A occurring.
  • P(B) is the probability of event B occurring.
  • P(A and B) is the probability of both event A and event B occurring.

Now, let's calculate each of these probabilities:

  1. P(A): Probability of rolling a multiple of 3 on the first roll.

    We found that there are 12 outcomes that satisfy Event A. Since there are 36 possible outcomes in total, the probability of Event A is:

    P(A) = 12 / 36 = 1 / 3

  2. P(B): Probability of rolling a total of 10 for both rolls.

    We found that there are 3 outcomes that satisfy Event B. Therefore, the probability of Event B is:

    P(B) = 3 / 36 = 1 / 12

  3. P(A and B): Probability of rolling a multiple of 3 on the first roll and a total of 10.

    We identified that there is 1 outcome (6, 4) that satisfies both events. So, the probability of both events occurring is:

    P(A and B) = 1 / 36

Now, we can plug these values into the Addition Rule formula:

P(A or B) = P(A) + P(B) - P(A and B) P(A or B) = (1 / 3) + (1 / 12) - (1 / 36)

To add these fractions, we need a common denominator, which is 36:

P(A or B) = (12 / 36) + (3 / 36) - (1 / 36) P(A or B) = (12 + 3 - 1) / 36 P(A or B) = 14 / 36

Finally, we can simplify the fraction:

P(A or B) = 7 / 18

Therefore, the probability of getting either a multiple of 3 on the first roll or a total of 10 for both rolls when a die is rolled twice is 7/18. This problem illustrates the importance of understanding basic probability principles, identifying favorable outcomes, and applying the Addition Rule of Probability. By breaking down the problem into smaller parts and systematically analyzing each part, we can arrive at the correct solution.

Understanding probability is not just an academic exercise; it has practical applications in various real-world scenarios. From assessing risks in finance to making informed decisions in everyday life, a solid grasp of probability concepts is invaluable. This article has provided a detailed walkthrough of a specific probability problem, but the principles and methods discussed can be applied to a wide range of similar problems. By practicing and applying these concepts, you can enhance your problem-solving skills and gain a deeper understanding of the world around you.

To solidify your understanding of probability, try solving similar problems. Here are a few suggestions:

  1. What is the probability of rolling an even number on the first roll or a total of 7 for both rolls?
  2. What is the probability of rolling a number greater than 4 on the first roll or a total less than 5 for both rolls?
  3. What is the probability of rolling the same number on both rolls or a total of 8?

By tackling these and other probability problems, you'll build confidence in your abilities and develop a stronger intuition for probability concepts. Remember, the key to mastering probability is practice and a systematic approach to problem-solving.