Probability Of Hitting Cracked Glass In A Window

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This article delves into a probability problem concerning a large window composed of 15 equally sized pieces of glass, where 3 pieces are cracked. We aim to determine the probability of a rock hitting a cracked piece of glass if thrown at the window. This problem falls under the realm of basic probability, a fundamental concept in mathematics and statistics. Understanding probability is crucial in various fields, from games of chance to risk assessment and scientific research. Before we dive into the solution, let's first define probability and its essential elements.

Probability is a numerical 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 higher the probability of an event, the more likely it is to happen. In this case, our event is the rock hitting a cracked piece of glass. To calculate probability, we need to consider the following:

  • Sample Space: The set of all possible outcomes of an experiment. In our scenario, the sample space consists of all 15 pieces of glass, as the rock could potentially hit any of them.

  • Event: A subset of the sample space that we are interested in. Here, our event is the rock hitting a cracked piece of glass, which comprises the 3 cracked pieces.

  • Probability Formula: The probability of an event is calculated by dividing the number of favorable outcomes (outcomes in the event) by the total number of possible outcomes (the size of the sample space). Mathematically, this is expressed as:

    Probability (Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

With this foundation in place, we can now proceed to solve the problem step by step, applying the probability formula to arrive at the solution. Understanding the basics of probability allows us to approach such problems methodically and arrive at accurate conclusions. This problem serves as a great example of how probability concepts can be applied to real-world scenarios, highlighting the practical relevance of this mathematical field.

Let's restate the problem clearly. Imagine a large window meticulously crafted from 15 individual pieces of glass. Each piece is identical in size and shape, forming a cohesive whole. However, the tranquility of this window is marred by a minor imperfection: 3 of these glass pieces have developed cracks. Now, picture a scenario where a rock is thrown haphazardly towards this window. Assuming the rock is equally likely to hit any part of the window, what is the probability that the rock will strike one of the cracked pieces of glass? This question invites us to apply the principles of probability to determine the likelihood of a specific event occurring within a defined sample space. The core of the problem lies in understanding the ratio between the number of cracked pieces and the total number of pieces, which will directly translate into the probability we seek.

To solve this, we need to identify the key elements: the total possible outcomes and the favorable outcomes. The total possible outcomes represent the entire sample space, which in this case is the 15 pieces of glass. The favorable outcomes are the instances we are interested in, which are the 3 cracked pieces. The probability will then be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This simple yet powerful calculation will reveal the likelihood of the rock hitting a cracked piece, providing a quantitative answer to our question. This problem highlights the practical application of probability in everyday scenarios, showcasing how mathematical concepts can help us understand and predict the likelihood of events.

To determine the probability that the rock hits a cracked piece of glass, we need to apply the basic probability formula, which states:

Probability (Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

In our scenario:

  • Total Number of Possible Outcomes: This is the total number of pieces of glass in the window, which is 15.
  • Number of Favorable Outcomes: This is the number of cracked pieces of glass, which is 3.

Now, we can plug these values into the formula:

Probability (Rock hits a cracked piece) = 3 / 15

This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3:

Probability (Rock hits a cracked piece) = (3 ÷ 3) / (15 ÷ 3) = 1 / 5

Therefore, the probability that the rock hits a cracked piece of glass is 1/5. This fraction can also be expressed as a decimal or a percentage. To convert it to a decimal, we divide 1 by 5, which gives us 0.2. To convert it to a percentage, we multiply the decimal by 100, which gives us 20%. Thus, there is a 20% chance that the rock will hit a cracked piece of glass. This calculation demonstrates a clear and concise application of probability principles to a practical problem, illustrating how we can quantify the likelihood of an event based on the number of favorable outcomes compared to the total possible outcomes.

The probability of the rock hitting a cracked piece of glass, as we calculated, is 1/5. While 1/5 is a perfectly valid and accurate way to express the probability, it can also be represented in other forms, such as a decimal or a percentage. Understanding these different representations allows for a more intuitive grasp of the likelihood of the event. We've already shown that the probability as a fraction is 1/5. Now, let's explore its decimal and percentage equivalents.

To convert the fraction 1/5 to a decimal, we simply perform the division: 1 divided by 5. The result is 0.2. Therefore, the probability can also be expressed as 0.2. This decimal representation provides a different perspective on the likelihood of the event. A probability of 0 indicates impossibility, while a probability of 1 indicates certainty. A value of 0.2 falls closer to impossibility, suggesting that it is less likely than not that the rock will hit a cracked piece of glass. However, it is still a tangible possibility, as it is not at the extreme end of the scale.

Next, let's convert the decimal 0.2 to a percentage. To do this, we multiply the decimal by 100. So, 0.2 multiplied by 100 equals 20. Therefore, the probability can also be expressed as 20%. This percentage representation is often the most intuitive for many people, as it provides a direct sense of the chance of the event occurring. A probability of 20% means that if we were to repeat the rock-throwing scenario many times, we would expect the rock to hit a cracked piece of glass approximately 20% of the time. This provides a tangible way to understand the likelihood of the event in a real-world context.

In summary, the probability of the rock hitting a cracked piece of glass can be expressed in three equivalent forms: the fraction 1/5, the decimal 0.2, and the percentage 20%. Each representation provides a slightly different perspective on the likelihood of the event, allowing for a more comprehensive understanding of the probability involved.

In conclusion, the probability that the rock hits a cracked piece of glass in the window is 1/5, which is equivalent to 0.2 or 20%. This result was obtained by applying the fundamental principles of probability, specifically by dividing the number of favorable outcomes (3 cracked pieces) by the total number of possible outcomes (15 pieces of glass). This problem serves as a clear illustration of how basic probability concepts can be used to analyze real-world scenarios and quantify the likelihood of specific events.

Throughout this article, we've explored the concept of probability, defined its key components such as sample space and events, and applied the probability formula to solve the problem at hand. We've also discussed how the probability can be expressed in different forms – as a fraction, a decimal, and a percentage – each offering a slightly different perspective on the likelihood of the event. Understanding these different representations is crucial for effectively communicating and interpreting probability in various contexts.

This problem, while seemingly simple, highlights the importance of probability in everyday life. From assessing risks to making informed decisions, probability plays a vital role in many aspects of our lives. By grasping the fundamental principles of probability, we can better understand the world around us and make more informed choices. This exercise demonstrates how mathematical concepts can be applied to practical situations, making mathematics a relevant and valuable tool for problem-solving and decision-making. The ability to calculate and interpret probabilities is a valuable skill that empowers us to navigate uncertainty and make predictions about future events.

Q: What is the formula for calculating probability?

A: The formula for calculating probability is: Probability (Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Q: How do you express probability as a percentage?

A: To express probability as a percentage, multiply the decimal form of the probability by 100.

Q: What does a probability of 0 mean?

A: A probability of 0 means that the event is impossible and will not occur.

Q: What does a probability of 1 mean?

A: A probability of 1 means that the event is certain to occur.

Q: Why is understanding probability important?

A: Understanding probability is important because it helps us assess risks, make informed decisions, and understand the likelihood of events in various situations, from games of chance to scientific research.