Probability Calculation For Tree Selection A Landscaper's Dilemma

In the realm of probability, we often encounter scenarios where we need to determine the likelihood of specific events occurring. This article delves into a problem involving a landscaper selecting trees, exploring the probability of choosing trees of two different types. We'll dissect the problem, analyze the possible outcomes, and arrive at the solution, expressing the probability as a percentage. This comprehensive exploration aims to provide a clear understanding of the problem-solving process and the underlying probabilistic concepts. Probability, as a branch of mathematics, deals with the chance of an event occurring. In this context, we'll calculate the likelihood of a landscaper selecting a mix of deciduous and evergreen trees from a given selection. The problem is not just a mathematical exercise; it mirrors real-world scenarios where decisions involve assessing probabilities, such as in resource allocation, risk management, or even in everyday choices. By understanding the principles at play, we can better navigate situations involving uncertainty and make more informed decisions. The following sections will guide you through the problem-solving steps, from identifying the key elements of the problem to calculating the final probability. So, let's embark on this journey of exploring probabilities and problem-solving, enhancing our understanding of how mathematics plays a crucial role in interpreting and predicting events around us. This problem provides a perfect example of how theoretical probability can be applied in practical scenarios, offering a blend of mathematical precision and real-world relevance.

Problem Statement

A landscaper is selecting two trees to plant. He has five trees to choose from, three of which are deciduous and two are evergreen. What is the probability that he chooses trees of two different types? Express your answer as a percent.

Solution

To solve this problem, we need to calculate the probability of the landscaper selecting one deciduous tree and one evergreen tree. We can approach this by first determining the total number of ways to choose two trees from the five available, and then calculating the number of ways to choose one deciduous and one evergreen tree. This approach allows us to systematically break down the problem and apply the fundamental principles of probability. Let's delve into the calculations to find the solution. The problem highlights the importance of understanding combinations in probability. A combination is a selection of items from a set where the order of selection does not matter. In this case, whether the landscaper picks a deciduous tree first or an evergreen tree first doesn't change the outcome – what matters is the final pair of trees chosen. This is a key concept in combinatorial mathematics and is essential for solving many probability problems. Understanding this concept is crucial for accurately calculating the probabilities involved in the given scenario. The problem effectively blends the principles of combinations with the calculation of probabilities, offering a comprehensive exercise in applying mathematical concepts to real-world scenarios. By solving this problem, we gain not only the answer but also a deeper appreciation for the elegance and utility of mathematical tools in analyzing and interpreting events.

1. Calculate the Total Number of Ways to Choose Two Trees

The total number of ways to choose two trees from five is given by the combination formula:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Where ${ n }$ is the total number of items, ${ k }$ is the number of items to choose, and ${ ! }$ denotes the factorial.

In this case, ${ n = 5 }$ (total trees) and ${ k = 2 }$ (trees to choose).

$$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)} = \frac{5 \times 4}{2 \times 1} = 10$$

So, there are 10 possible ways to choose two trees from the five trees. This calculation forms the foundation for determining the probability. By understanding the total possible outcomes, we can then focus on the specific outcomes that meet our criteria – in this case, selecting one deciduous and one evergreen tree. The use of the combination formula is a fundamental tool in probability and combinatorics, allowing us to systematically count the number of ways to choose items from a larger set. This step is crucial for accurately determining the probability, as it provides the denominator in the probability fraction. Without knowing the total possible outcomes, it would be impossible to calculate the likelihood of a specific event occurring. This underscores the importance of mastering combinatorial principles for solving probability problems. The initial calculation sets the stage for the subsequent steps, highlighting the interconnectedness of mathematical concepts in problem-solving.

2. Calculate the Number of Ways to Choose One Deciduous and One Evergreen Tree

There are three deciduous trees and two evergreen trees. To choose one of each type, we multiply the number of ways to choose one deciduous tree from three by the number of ways to choose one evergreen tree from two.

Number of ways to choose one deciduous tree from three: ${ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3!}{1!2!} = \frac{3 \times 2 \times 1}{(1)(2 \times 1)} = 3 }$

Number of ways to choose one evergreen tree from two: ${ \binom{2}{1} = \frac{2!}{1!(2-1)!} = \frac{2!}{1!1!} = \frac{2 \times 1}{(1)(1)} = 2 }$

So, the number of ways to choose one deciduous and one evergreen tree is:

$$3 \times 2 = 6$$

This step narrows our focus from the total possible outcomes to the specific outcomes that satisfy the problem's condition. By calculating the number of ways to choose one tree from each category, we are essentially quantifying the favorable outcomes. This is a crucial step in determining the probability, as it provides the numerator in the probability fraction. The multiplication principle used here is a fundamental concept in combinatorics, stating that if there are ${ m }$ ways to do one thing and ${ n }$ ways to do another, then there are ${ m \times n }$ ways to do both. This principle is widely applicable in various scenarios, from counting the number of possible passwords to calculating the number of different meal combinations in a restaurant. The application of the multiplication principle in this context demonstrates its versatility and importance in problem-solving. This calculation bridges the gap between the individual probabilities of selecting each type of tree and the combined probability of selecting both types, showcasing the elegance of mathematical reasoning.

3. Calculate the Probability

The probability of choosing trees of two different types is the ratio of the number of ways to choose one deciduous and one evergreen tree to the total number of ways to choose two trees:

$$P(\text{two different types}) = \frac{\text{Number of ways to choose one deciduous and one evergreen tree}}{\text{Total number of ways to choose two trees}} = \frac{6}{10} = \frac{3}{5}$$

4. Express the Probability as a Percentage

To express the probability as a percentage, multiply the fraction by 100:

$$\frac{3}{5} \times 100 = 60\%$$

Thus, the probability that the landscaper chooses trees of two different types is 60%. This final calculation converts the probability from a fraction to a percentage, making it easier to interpret and understand. Probabilities are often expressed as percentages in real-world scenarios to convey the likelihood of an event in a more intuitive way. A probability of 60% indicates a relatively high chance of the event occurring, suggesting that the landscaper is more likely to choose a mix of deciduous and evergreen trees than to choose two trees of the same type. This conversion underscores the importance of being able to express probabilities in various forms, depending on the context and the audience. The final answer not only solves the problem but also provides a valuable piece of information that can be used for decision-making. This highlights the practical relevance of probability calculations in everyday life and in various professional fields.

Answer

The probability that the landscaper chooses trees of two different types is 60%. Therefore, the correct answer is not among the options provided (A. 30%, B. 40%). The correct answer should be 60%.

Conclusion

In this article, we've walked through the process of calculating the probability of a landscaper selecting two trees of different types from a set of five trees, three deciduous and two evergreen. We used the principles of combinations to determine the total possible outcomes and the favorable outcomes, and then calculated the probability as a percentage. The problem illustrates the practical application of probability and combinatorics in real-world scenarios. By systematically breaking down the problem and applying the appropriate mathematical tools, we were able to arrive at the correct solution. This problem is not just an isolated exercise in mathematics; it reflects the broader applicability of probability in various fields, from statistics and finance to engineering and decision-making. Understanding probability allows us to quantify uncertainty and make informed decisions in situations where outcomes are not certain. The skills developed in solving this problem, such as identifying key information, applying mathematical formulas, and interpreting results, are transferable to many other problem-solving contexts. This reinforces the importance of mastering mathematical concepts and their applications in the real world. The journey through this problem has not only provided us with an answer but also enhanced our problem-solving abilities and our appreciation for the role of mathematics in interpreting and predicting events around us. The ability to calculate and interpret probabilities is a valuable skill in today's data-driven world, empowering us to make more informed choices and better understand the world around us. Thus, the problem serves as a microcosm of the larger landscape of mathematical applications, highlighting the power and versatility of mathematical thinking. The knowledge and skills gained from this problem can be applied to a wide range of scenarios, making it a valuable addition to our problem-solving toolkit. The journey through this problem demonstrates the power of mathematical reasoning and its ability to illuminate real-world situations.

Keywords

Probability, combinations, deciduous trees, evergreen trees, landscaper, percentage, problem-solving, mathematics, tree selection, total outcomes, favorable outcomes.