Probability Of Lighting A Room With Faulty Bulbs

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At the heart of this problem lies the concept of probability, a fundamental aspect of mathematics that helps us quantify the likelihood of events. The scenario presented involves a room with three lamp sockets and a selection of ten bulbs, some faulty and some functional. Our goal is to determine the probability that the room will be lighted when three bulbs are randomly chosen and placed in the sockets. This seemingly simple problem touches upon crucial principles of combinatorics and probability theory. To unravel this, we will explore the different ways to select bulbs, the combinations that lead to a lit room, and ultimately, the probability of achieving this outcome. The problem provides a practical application of probability in everyday scenarios, highlighting the importance of understanding how random events can influence outcomes. This exploration will not only provide the answer but also shed light on the thought process involved in solving probability problems. In probability theory, we frequently deal with scenarios where the outcome is uncertain but can be quantified. This problem beautifully illustrates such a scenario, requiring us to think critically about the conditions under which the room lights up and the various combinations of bulbs that can be selected. The challenge encourages us to break down the problem into manageable parts, calculate individual probabilities, and combine them to arrive at the final solution. The ability to solve such problems is not only valuable in mathematics but also in various real-world applications, where decision-making often involves assessing probabilities and weighing potential outcomes.

The problem presents a scenario where a room has three lamp sockets, and there's a collection of 10 bulbs. Among these bulbs, 6 are faulty, meaning they won't light up, and the remaining 4 are functional. A person randomly selects 3 bulbs from this collection and places them into the sockets. The question we aim to answer is: What is the probability that the room will be lighted? This implies that at least one of the selected bulbs must be functional for the room to light up. To solve this, we need to consider the total number of ways to choose 3 bulbs from 10 and then determine the number of ways to choose 3 bulbs such that at least one is functional. The probability will then be the ratio of the favorable outcomes (at least one functional bulb) to the total possible outcomes (any 3 bulbs). Understanding this problem requires a clear grasp of combinations and how to calculate probabilities in scenarios with multiple possible outcomes. It's a classic example of how probability theory can be applied to analyze real-world situations. The key to solving this problem lies in carefully considering the different scenarios that can lead to the room being lit and then calculating the probabilities associated with those scenarios. We must account for the fact that selecting one, two, or three functional bulbs will all result in the room lighting up. The problem highlights the importance of thinking systematically about all possible outcomes and their respective probabilities.

To solve this probability problem effectively, we need to break it down into smaller, manageable steps. This approach allows us to tackle each aspect of the problem systematically and ensures that we don't miss any crucial details. Here's a step-by-step analysis:

  1. Identify the Total Possible Outcomes: First, we need to determine the total number of ways to select 3 bulbs from a collection of 10. This is a combination problem, as the order in which the bulbs are selected doesn't matter. We'll use the combination formula to calculate this.
  2. Identify the Unfavorable Outcomes: Next, we need to determine the number of ways to select 3 bulbs such that the room won't be lighted. This means we need to select 3 faulty bulbs from the 6 faulty bulbs available. Again, we'll use the combination formula.
  3. Calculate the Number of Favorable Outcomes: The number of favorable outcomes (the room being lighted) is the difference between the total possible outcomes and the unfavorable outcomes. This is because the favorable outcomes are all the scenarios where we don't select 3 faulty bulbs.
  4. Calculate the Probability: Finally, we calculate the probability of the room being lighted by dividing the number of favorable outcomes by the total number of possible outcomes. This gives us the probability that at least one of the selected bulbs is functional.

By following these steps, we can systematically approach the problem and arrive at the correct solution. Each step involves careful consideration of the combinations involved and how they contribute to the overall probability. This step-by-step approach not only helps in solving this specific problem but also provides a general framework for tackling other probability problems. It emphasizes the importance of breaking down complex problems into simpler parts and addressing each part individually.

Now, let's delve into the calculations to find the probability of the room being lighted. We'll follow the steps outlined in the previous section:

  1. Total Possible Outcomes: The total number of ways to select 3 bulbs from 10 is given by the combination formula:

    C(n, k) = n! / (k!(n-k)!) 
    

    where n is the total number of items, k is the number of items to choose, and ! denotes the factorial.

    So, C(10, 3) = 10! / (3!7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120.

    There are 120 total possible ways to select 3 bulbs from 10.

  2. Unfavorable Outcomes: The number of ways to select 3 faulty bulbs from 6 faulty bulbs is:

    C(6, 3) = 6! / (3!3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20.

    There are 20 ways to select 3 faulty bulbs, which means the room won't be lighted.

  3. Favorable Outcomes: The number of ways to select at least one functional bulb (favorable outcomes) is the total outcomes minus the unfavorable outcomes:

    Favorable Outcomes = Total Outcomes - Unfavorable Outcomes = 120 - 20 = 100.

    There are 100 ways to select 3 bulbs such that at least one is functional.

  4. Probability Calculation: The probability of the room being lighted is the number of favorable outcomes divided by the total outcomes:

    Probability = Favorable Outcomes / Total Outcomes = 100 / 120 = 5/6.

Therefore, the probability that the room will be lighted is 5/6. This calculation demonstrates how we can use combinations and probability theory to solve real-world problems. By breaking down the problem into smaller steps and calculating each part systematically, we arrive at the final answer. The use of the combination formula is crucial in this calculation, as it allows us to account for the different ways to select bulbs without considering the order of selection.

Based on our calculations, the probability that the room will be lighted when 3 bulbs are randomly selected from the collection of 10 (6 faulty and 4 functional) is 5/6. This result indicates a high likelihood that the room will be lighted, as there's a greater chance of selecting at least one functional bulb compared to selecting three faulty bulbs. The explanation for this probability lies in the ratio of functional bulbs to faulty bulbs in the collection. With 4 out of 10 bulbs being functional, there's a relatively high chance that any randomly selected bulb will be functional. When selecting 3 bulbs, the probability of all three being faulty is significantly lower than the probability of at least one being functional. To further illustrate this, consider the complementary probability: the probability that the room will not be lighted. This is the probability of selecting 3 faulty bulbs, which we calculated earlier as 20 out of 120 total outcomes, or 1/6. Since the probability of the room being lighted and the probability of the room not being lighted must add up to 1, we can also calculate the probability of the room being lighted as 1 - (1/6) = 5/6. This approach provides an alternative way to verify our initial calculation. The result highlights the importance of considering both favorable and unfavorable outcomes when calculating probabilities. By understanding the relationship between these outcomes, we can gain a deeper insight into the likelihood of different events occurring.

While we've solved the problem using a step-by-step approach involving combinations and probability calculations, it's beneficial to explore alternative methods to gain a more comprehensive understanding. Here are a couple of alternative approaches:

  1. Complementary Probability: As mentioned earlier, we can use the concept of complementary probability. Instead of directly calculating the probability of the room being lighted, we calculate the probability of the room not being lighted (i.e., selecting 3 faulty bulbs) and subtract it from 1. This approach can be simpler in some cases, especially when calculating the probability of the complement is easier than calculating the probability of the event directly.

    • Probability (room not lighted) = Probability (selecting 3 faulty bulbs) = C(6, 3) / C(10, 3) = 20 / 120 = 1/6
    • Probability (room lighted) = 1 - Probability (room not lighted) = 1 - 1/6 = 5/6
  2. Direct Calculation of Favorable Outcomes: We can directly calculate the number of ways to select at least one functional bulb by considering the different scenarios:

    • Selecting 1 functional bulb and 2 faulty bulbs: C(4, 1) * C(6, 2) = 4 * 15 = 60
    • Selecting 2 functional bulbs and 1 faulty bulb: C(4, 2) * C(6, 1) = 6 * 6 = 36
    • Selecting 3 functional bulbs: C(4, 3) = 4
    • Total favorable outcomes = 60 + 36 + 4 = 100
    • Probability (room lighted) = Total favorable outcomes / Total outcomes = 100 / 120 = 5/6

These alternative approaches demonstrate that there isn't always a single way to solve a probability problem. By exploring different methods, we can not only verify our results but also deepen our understanding of the underlying concepts. The complementary probability approach provides a useful shortcut in certain scenarios, while the direct calculation approach reinforces the understanding of how different combinations contribute to the final probability.

The problem of calculating the probability of lighting a room with faulty bulbs might seem like a purely theoretical exercise, but it highlights the relevance of probability in everyday life. Probability concepts are applied in various fields, from simple decision-making to complex scientific research. Here are a few examples:

  1. Quality Control: In manufacturing, probability is used to assess the likelihood of producing defective items. For instance, a factory might use probability to determine the chance of a batch of products containing a certain number of faulty items. This helps in implementing quality control measures and ensuring product reliability.
  2. Risk Assessment: In finance and insurance, probability is crucial for assessing risks. Insurers use probability to calculate the likelihood of events like accidents or natural disasters, which helps them determine insurance premiums. Financial analysts use probability to evaluate investment risks and make informed decisions.
  3. Medical Diagnosis: Doctors use probability to assess the likelihood of a patient having a particular disease based on symptoms and test results. Understanding probabilities helps in making accurate diagnoses and choosing appropriate treatments.
  4. Weather Forecasting: Weather forecasts are based on probability. Meteorologists use models to predict the likelihood of rain, snow, or other weather events. These probabilities help people plan their activities and prepare for different weather conditions.
  5. Games of Chance: Probability is the foundation of games of chance, such as lotteries, card games, and casino games. Understanding the probabilities involved can help players make informed decisions and avoid being misled by luck.

These examples illustrate that probability is not just an abstract mathematical concept but a powerful tool for understanding and navigating the uncertainties of the world around us. By developing a strong understanding of probability, we can make better decisions and better assess the risks and opportunities that we encounter in our daily lives.

In conclusion, the problem of determining the probability of lighting a room with faulty bulbs serves as an excellent illustration of how probability theory can be applied to real-world scenarios. By systematically breaking down the problem, calculating combinations, and considering both favorable and unfavorable outcomes, we arrived at the solution of 5/6. This probability indicates a high likelihood that the room will be lighted, given the ratio of functional to faulty bulbs. We also explored alternative approaches to solving the problem, such as using complementary probability and directly calculating favorable outcomes, which reinforced our understanding of the underlying concepts. Furthermore, we discussed the wide-ranging applications of probability in everyday life, highlighting its importance in fields like quality control, risk assessment, medical diagnosis, and weather forecasting. Mastering probability is not only valuable for academic pursuits but also for making informed decisions in various aspects of life. By developing a strong understanding of probability, we can better assess risks, evaluate opportunities, and navigate the uncertainties of the world around us. This problem, while seemingly simple, encapsulates the essence of probability theory and its practical applications. The ability to solve such problems demonstrates a solid foundation in probability concepts and the ability to apply them effectively. As we continue to encounter situations involving uncertainty, a firm grasp of probability will undoubtedly prove to be a valuable asset.