Probability Calculation Boy Cycles Given Lateness

In probability theory, we often encounter scenarios where we need to determine the likelihood of an event occurring given that another event has already happened. This is where the concept of conditional probability comes into play. Conditional probability is a fundamental concept in statistics and probability theory that helps us understand the relationship between events. The problem presented, where a boy walks or cycles to school and is sometimes late, perfectly illustrates this concept. We are given the probabilities of the boy walking and cycling, as well as the probabilities of him being late given each mode of transportation. Our task is to find the probability that he cycled to school given that he was late. This type of problem requires us to use Bayes' Theorem, a powerful tool for calculating conditional probabilities. Understanding these probabilities can provide valuable insights into the factors influencing the boy's punctuality. Bayes' Theorem is not just a mathematical formula; it's a powerful tool with applications in various fields, including machine learning, medical diagnosis, and risk assessment. In essence, it allows us to update our beliefs about an event based on new evidence. By understanding the probabilities of the boy's mode of transportation and his lateness, we can make informed decisions and predictions. This detailed exploration will walk you through the steps involved in solving this problem, providing a clear understanding of conditional probability and its applications. This involves breaking down the problem into smaller, manageable parts, identifying the relevant probabilities, and applying Bayes' Theorem to arrive at the solution. By the end of this exploration, you will have a solid grasp of how to tackle similar probability problems and understand the underlying principles that govern them. This problem is a classic example of how probability theory can be used to model real-world situations and make predictions based on available data. So, let's dive into the problem and explore the fascinating world of conditional probability.

A boy walks to school 60% of the time and cycles 40% of the time. He is late to school 5% of the times that he walks, and he is late 2% of the times that he cycles. Given that he is late to school, find the probability that he cycled. This problem presents a classic scenario for applying conditional probability. We're given the probabilities of two primary events: the boy walking (W) and the boy cycling (C). Additionally, we have information about the conditional probabilities of him being late (L) given that he walked (P(L|W)) and given that he cycled (P(L|C)). The core question is to determine the reverse conditional probability: P(C|L), which is the probability that he cycled given that he is late. The given percentages can be directly converted into probabilities. For instance, walking 60% of the time translates to a probability of 0.6, and cycling 40% of the time means a probability of 0.4. Similarly, the lateness probabilities can be expressed as decimals: 5% late when walking becomes 0.05, and 2% late when cycling becomes 0.02. The problem requires careful consideration of the relationships between these probabilities. We need to account for both the likelihood of cycling and the likelihood of being late when cycling. This involves using a formula that combines these probabilities to give us the desired conditional probability. The key to solving this problem lies in recognizing the need for Bayes' Theorem or a similar approach to calculate conditional probabilities. Understanding the nuances of this theorem is crucial for accurately determining the probability of cycling given lateness. The problem statement provides all the necessary information to solve for P(C|L). By carefully organizing the given probabilities and applying the correct formula, we can arrive at the solution. This problem is not only a mathematical exercise but also a practical example of how probability theory can be applied to real-world scenarios. It demonstrates the importance of considering conditional probabilities when making inferences and predictions. Therefore, a clear understanding of the problem statement and the given information is the first step towards finding the correct solution. The next step is to translate these pieces of information into mathematical terms and then apply the appropriate formula to calculate the desired probability.

To solve this probability problem, let's first define the events and their corresponding probabilities. It's crucial to translate the word problem into a clear mathematical representation. Let's define the following events:

• W: The boy walks to school.
• C: The boy cycles to school.
• L: The boy is late to school.

From the problem statement, we are given the following probabilities:

• P(W) = 0.60 (Probability of walking)
• P(C) = 0.40 (Probability of cycling)
• P(L|W) = 0.05 (Probability of being late given he walks)
• P(L|C) = 0.02 (Probability of being late given he cycles)

Our goal is to find P(C|L), the probability that the boy cycled given that he is late. Clearly defining these events and probabilities is a critical first step in solving any probability problem. It helps us to organize the information and avoid confusion. By assigning symbols to each event and converting the percentages into decimal probabilities, we create a solid foundation for applying the relevant formulas and theorems. The probabilities P(W) and P(C) represent the prior probabilities of the boy walking or cycling, respectively. These are the probabilities before we have any information about whether he is late. The probabilities P(L|W) and P(L|C) represent the conditional probabilities of being late given each mode of transportation. These are the probabilities that he is late, knowing that he walked or cycled. The probability we are trying to find, P(C|L), is the posterior probability. It represents our updated belief about the likelihood of him cycling, given the new information that he is late. This distinction between prior and posterior probabilities is at the heart of Bayesian reasoning. Understanding the notation and what each probability represents is crucial for correctly applying Bayes' Theorem or other probability rules. It's like having a map before embarking on a journey; a clear definition of events and probabilities guides us towards the solution. With these definitions in place, we can now proceed to the next step, which involves applying the appropriate formula to calculate P(C|L).

To find P(C|L), the probability that the boy cycled given that he is late, we can use Bayes' Theorem. Bayes' Theorem is a fundamental principle in probability theory that describes how to update the probability of a hypothesis based on new evidence. In this case, our hypothesis is that the boy cycled, and the evidence is that he is late. Bayes' Theorem is particularly useful when we want to reverse the direction of a conditional probability. That is, we know P(L|C) (the probability of being late given he cycles) and we want to find P(C|L) (the probability he cycles given he is late). The formula for Bayes' Theorem is as follows:

P(C|L) = [P(L|C) * P(C)] / P(L)

Where:

• P(C|L) is the probability of cycling given he is late (what we want to find).
• P(L|C) is the probability of being late given he cycles (0.02).
• P(C) is the probability of cycling (0.40).
• P(L) is the overall probability of being late.

We already have P(L|C) and P(C). However, we need to calculate P(L), the overall probability of being late. To find P(L), we can use the law of total probability. The law of total probability states that the probability of an event can be calculated by summing the probabilities of that event occurring under each possible condition. In this case, the boy can be late either by walking or by cycling. So, we can calculate P(L) as follows:

P(L) = P(L|W) * P(W) + P(L|C) * P(C)

Substituting the values we have:

P(L) = (0.05 * 0.60) + (0.02 * 0.40)

Calculating this gives us:

P(L) = 0.03 + 0.008 = 0.038

Now that we have P(L), we can substitute all the values into Bayes' Theorem:

P(C|L) = (0.02 * 0.40) / 0.038

Bayes' Theorem is a cornerstone of probabilistic reasoning and has applications in diverse fields, from medical diagnosis to machine learning. It allows us to update our beliefs in light of new evidence, making it a powerful tool for decision-making under uncertainty. In this problem, it helps us to determine the probability that the boy cycled given the information that he is late. The law of total probability is another important concept that complements Bayes' Theorem. It allows us to calculate the overall probability of an event by considering all the possible ways it can occur. In this case, the boy can be late either by walking or cycling, and the law of total probability helps us to combine these probabilities to find the overall probability of being late. The calculation of P(L) is a crucial step in applying Bayes' Theorem. Without P(L), we cannot determine the posterior probability P(C|L). The formula for P(L) ensures that we are accounting for all the ways the event L (being late) can occur, given the possible modes of transportation. Once we have calculated P(L), we can plug it into Bayes' Theorem along with the other known probabilities to find the desired conditional probability. This step-by-step approach ensures that we are following the correct logical sequence and arriving at the accurate solution. Therefore, a clear understanding of Bayes' Theorem and the law of total probability is essential for solving problems of this nature. These concepts provide a robust framework for reasoning about probabilities and making informed decisions in the face of uncertainty.

Now that we have all the necessary components, let's perform the calculation to find P(C|L). From the previous section, we have:

P(L) = 0.038

And Bayes' Theorem is:

P(C|L) = [P(L|C) * P(C)] / P(L)

Substituting the values:

P(C|L) = (0.02 * 0.40) / 0.038

Performing the multiplication in the numerator:

P(C|L) = 0.008 / 0.038

Now, divide 0.008 by 0.038:

P(C|L) ≈ 0.2105

Therefore, the probability that the boy cycled given that he is late is approximately 0.2105, or 21.05%. This means that out of all the times the boy is late, about 21.05% of those times he cycled to school. The final calculation involves a simple division, but it's crucial to ensure that the preceding steps have been performed accurately. Any error in calculating P(L) or in the initial probabilities will propagate through the calculation and lead to an incorrect result. The result, 0.2105, is a probability, so it must be a value between 0 and 1. This serves as a quick check to ensure that the answer is reasonable. Converting the probability to a percentage, 21.05%, helps to provide a more intuitive understanding of the result. It tells us the proportion of times the boy cycled when he was late. This probability is less than 0.5, which means that it's more likely that he walked when he was late. This makes sense given that he walks more often (60% of the time) and is also more likely to be late when walking (5% compared to 2% when cycling). The interpretation of the result is just as important as the calculation itself. It helps us to understand the practical implications of the probability we have calculated. In this case, it tells us something about the relationship between the boy's mode of transportation and his lateness. This problem highlights the power of probability theory in analyzing real-world scenarios and drawing meaningful conclusions from data. By carefully applying the concepts of conditional probability and Bayes' Theorem, we can arrive at solutions that provide valuable insights. Therefore, the final solution to the problem is that the probability the boy cycled, given he is late, is approximately 0.2105 or 21.05%.

In conclusion, we have successfully calculated the probability that the boy cycled to school given that he was late. By carefully defining the events, applying Bayes' Theorem, and using the law of total probability, we found that the probability P(C|L) is approximately 0.2105, or 21.05%. This problem illustrates the power of conditional probability in analyzing real-world scenarios. It demonstrates how we can update our beliefs about an event based on new evidence. In this case, the evidence of the boy being late changes our belief about the likelihood that he cycled to school. The application of Bayes' Theorem is not limited to this specific problem. It has broad applications in various fields, including statistics, machine learning, and decision theory. Understanding Bayes' Theorem allows us to make more informed decisions in situations where uncertainty exists. The key to solving probability problems like this is to break them down into smaller, manageable steps. This involves clearly defining the events, identifying the relevant probabilities, and applying the appropriate formulas. By following a systematic approach, we can avoid confusion and arrive at the correct solution. The problem also highlights the importance of understanding the relationship between different probabilities. The conditional probabilities P(L|W) and P(L|C) provide information about the likelihood of being late given the mode of transportation. By combining these with the prior probabilities P(W) and P(C), we can calculate the posterior probability P(C|L). This process of updating probabilities based on new information is a fundamental aspect of Bayesian reasoning. Moreover, the use of the law of total probability to calculate P(L) is a crucial step in applying Bayes' Theorem. It ensures that we are considering all possible ways the event of being late can occur. The entire problem-solving process, from defining events to applying Bayes' Theorem and interpreting the result, provides a valuable learning experience in probability theory. It reinforces the importance of understanding the underlying concepts and applying them in a logical and systematic manner. Therefore, this problem serves as a excellent example of how probability theory can be used to model and analyze real-world situations, leading to insightful conclusions and informed decision-making.