What Is The Probability Of Getting A Specific Number In At Least One Of Three Throws Of A Six-sided Die?

Understanding probability is crucial in many areas, from everyday decision-making to complex scientific calculations. This article delves into a specific probability problem: determining the likelihood of rolling a particular number on at least one of three throws of a six-sided die. This seemingly simple question touches on fundamental concepts of probability, including independent events, complementary probabilities, and the application of combinatorial principles. We'll explore a step-by-step solution to this problem, providing clear explanations and insights that will not only help you solve this particular question but also enhance your overall understanding of probability calculations. By the end of this article, you'll be equipped with the knowledge to tackle similar probability challenges with confidence. So, let's embark on this journey to unravel the intricacies of rolling dice and calculating probabilities.

Let's consider the question: What is the probability of obtaining a specific number on at least one of the three rolls of a standard six-sided die? The provided options are: (A) 1/48, (B) 1/18, (C) 1/36, (D) 1/60, and (E) 1/24. This problem is a classic example of probability calculations, involving multiple independent events. To solve it effectively, we need to understand the underlying principles of probability, such as calculating the probability of an event not occurring and using the concept of complementary probability. The problem requires us to determine the chances of a successful outcome (rolling the specific number) occurring at least once in a series of trials (three rolls). This involves considering various scenarios, including the number appearing on the first roll, the second roll, the third roll, or any combination thereof. By carefully analyzing these scenarios and applying the appropriate formulas, we can arrive at the correct answer. The question challenges us to think critically about probability and apply our knowledge to a real-world scenario, making it an excellent exercise for anyone looking to improve their understanding of this mathematical field. Before diving into the solution, let's recap some essential probability concepts that will guide us through the process.

Before tackling the problem, let's solidify our understanding of fundamental probability concepts. Probability, at its core, 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. For instance, the probability of flipping a fair coin and landing on heads is 0.5, or 50%. This simple example illustrates the basic principle: Probability = (Number of favorable outcomes) / (Total number of possible outcomes). In the case of a six-sided die, each face (1 to 6) has an equal chance of appearing, so the probability of rolling any specific number is 1/6. This is because there is one favorable outcome (the specific number we want) and six possible outcomes (the numbers 1 through 6). When dealing with multiple events, we need to consider whether these events are independent or dependent. Independent events are those where the outcome of one does not affect the outcome of another. Rolling a die multiple times is an example of independent events; the result of one roll does not influence the result of the next roll. For independent events, we can calculate the probability of multiple events occurring by multiplying their individual probabilities. For example, the probability of rolling a 6 twice in a row is (1/6) * (1/6) = 1/36. Another crucial concept is complementary probability. The complementary probability of an event is the probability that the event will not occur. It is calculated as 1 minus the probability of the event occurring. This concept is particularly useful when calculating the probability of an event happening "at least once," as we'll see in the solution to our problem. With these fundamental concepts in mind, we are well-prepared to tackle the challenge of calculating the probability of rolling a specific number on a die multiple times.

To determine the probability of rolling a specific number on at least one of the three rolls, it's often easier to calculate the complementary probability first: the probability of not rolling the specific number on any of the three rolls. Then, we can subtract this complementary probability from 1 to find our desired probability. Let's break this down step by step. The probability of not rolling a specific number on a single roll of a six-sided die is 5/6. This is because there are five outcomes that are not the specific number we are interested in (if we're looking for a 1, the other outcomes are 2, 3, 4, 5, and 6) out of a total of six possible outcomes. Since the rolls are independent events, we can multiply the probabilities of not rolling the specific number on each of the three rolls together. This gives us (5/6) * (5/6) * (5/6) = 125/216. This fraction represents the probability of not rolling the specific number on any of the three rolls. Now, to find the probability of rolling the specific number at least once, we subtract the probability of not rolling it at all from 1. So, the calculation is 1 - (125/216). To perform this subtraction, we need to express 1 as a fraction with the same denominator, which is 216/216. Thus, the calculation becomes 216/216 - 125/216 = 91/216. Therefore, the probability of rolling a specific number on at least one of the three rolls of a six-sided die is 91/216. This result doesn't directly match any of the provided options (A) 1/48, (B) 1/18, (C) 1/36, (D) 1/60, (E) 1/24, indicating that either there may be a mistake in the options or the question might be misleading. However, this step-by-step solution demonstrates the correct approach to solving this type of probability problem. The key takeaway is the use of complementary probability to simplify the calculation, a technique that is widely applicable in various probability scenarios. While we have arrived at the correct probability of 91/216, it's essential to examine the given options and the context of the problem to ensure there are no discrepancies or alternative interpretations.

Having calculated the probability of rolling a specific number on at least one of the three rolls as 91/216, we now need to compare this result with the given options: (A) 1/48, (B) 1/18, (C) 1/36, (D) 1/60, and (E) 1/24. It's immediately apparent that our calculated probability, 91/216, does not directly match any of the provided options. This discrepancy raises a few possibilities. First, there might be an error in the options themselves. It's not uncommon for multiple-choice questions to contain incorrect answer choices. Second, there could be a misunderstanding or misinterpretation of the problem statement. While we have followed a logical and standard approach to solving this type of probability question, it's always prudent to double-check the problem's wording and assumptions. Third, the question might be designed to test a different aspect of probability, perhaps involving approximations or estimations. However, given the precise nature of probability calculations, it's unlikely that an approximation would be the intended solution. To further analyze the situation, let's convert our calculated probability, 91/216, into a decimal for easier comparison. Dividing 91 by 216 gives us approximately 0.421. Now, let's convert the options into decimals as well: (A) 1/48 ≈ 0.021, (B) 1/18 ≈ 0.056, (C) 1/36 ≈ 0.028, (D) 1/60 ≈ 0.017, and (E) 1/24 ≈ 0.042. Comparing these decimal values, it's clear that our calculated probability (0.421) is significantly higher than any of the options. This reinforces the conclusion that either the options are incorrect or there's a potential issue with the question itself. In such situations, it's crucial to review the steps taken in the calculation, ensuring that each step is logically sound and mathematically accurate. If the calculation is correct, the next step would be to consider the possibility of an error in the question or the answer choices.

In conclusion, we embarked on a journey to determine the probability of obtaining a specific number on at least one of the three rolls of a six-sided die. By applying fundamental probability principles, we calculated the probability to be 91/216, which is approximately 0.421. This involved calculating the complementary probability of not rolling the specific number on any of the three rolls and subtracting it from 1. However, upon comparing our calculated result with the provided options (A) 1/48, (B) 1/18, (C) 1/36, (D) 1/60, and (E) 1/24, we found a significant discrepancy. None of the options matched our calculated probability, suggesting a potential issue with either the options themselves or the question statement. This highlights the importance of not only understanding the mathematical concepts but also critically evaluating the results and the context of the problem. While we have successfully solved the problem using a standard probability approach, the mismatch with the given options underscores the need for careful review and consideration of all possibilities. This exercise serves as a valuable reminder that problem-solving often involves more than just arriving at a numerical answer; it also entails analyzing the validity and reasonableness of the solution in the given context. In practical scenarios, such discrepancies might prompt further investigation, clarification of the question, or re-evaluation of the available information. Ultimately, this exploration reinforces the multifaceted nature of probability and its application in real-world problem-solving.