Calculating The Probability Of Selecting A Two-Digit Cube Number
In mathematics, probability plays a crucial role in understanding the likelihood of events occurring. Calculating probabilities involves determining the ratio of favorable outcomes to the total possible outcomes. In this article, we will explore the process of calculating the probability of selecting a two-digit number that is also the cube of a natural number. This involves identifying the relevant set of numbers and then applying the fundamental principles of probability.
To begin, let's define the scope of our problem. We are interested in finding the probability of selecting a number from the set of two-digit natural numbers that is also a perfect cube. This means we need to identify the two-digit numbers that can be expressed as the cube of an integer. The set of two-digit natural numbers ranges from 10 to 99, inclusive. Our task is to determine how many of these numbers are perfect cubes and then calculate the probability of selecting one such number at random.
Before we dive into calculations, it's essential to clarify the concepts involved. A natural number is a positive integer (1, 2, 3, ...). A perfect cube is a number that can be obtained by cubing an integer (e.g., 8 is a perfect cube because 2^3 = 8). To solve this problem, we need to:
- Identify the range of two-digit natural numbers.
- Determine which natural numbers, when cubed, fall within this range.
- Calculate the total number of two-digit natural numbers.
- Calculate the number of two-digit natural numbers that are perfect cubes.
- Apply the probability formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes).
By systematically addressing each of these steps, we can accurately calculate the probability of selecting a two-digit cube number.
To identify two-digit perfect cubes, we need to find natural numbers whose cubes fall between 10 and 99. We can start by listing the cubes of natural numbers:
- 1^3 = 1
- 2^3 = 8
- 3^3 = 27
- 4^3 = 64
- 5^3 = 125
From this list, we can see that 3^3 = 27 and 4^3 = 64 are the only two-digit numbers that are perfect cubes. The cube of 2 (8) is a single-digit number, and the cube of 5 (125) is a three-digit number. Therefore, the two-digit perfect cubes are 27 and 64. This step is crucial as it narrows down the favorable outcomes for our probability calculation.
Now that we have identified the perfect cubes, we can move on to determining the total number of two-digit numbers, which will serve as the denominator in our probability fraction. This involves a straightforward calculation but is essential for the accurate determination of the final probability.
Next, we need to calculate the total number of two-digit numbers. Two-digit numbers range from 10 to 99, inclusive. To find the total count, we subtract the smallest two-digit number (10) from the largest (99) and add 1:
Total two-digit numbers = 99 - 10 + 1 = 90
There are 90 two-digit numbers in total. This value represents the total number of possible outcomes when selecting a number from the set of two-digit numbers. Understanding the total possible outcomes is crucial for calculating probability, as it forms the denominator of the probability fraction. With this information, we can now proceed to calculate the probability of selecting a two-digit cube number.
To determine the probability of selecting a two-digit cube number, we use the formula:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
We have already identified that there are 2 favorable outcomes (27 and 64) and 90 total possible outcomes (two-digit numbers from 10 to 99). Plugging these values into the formula:
Probability = 2 / 90
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
Probability = 1 / 45
Therefore, the probability of selecting a two-digit number that is a perfect cube is 1/45. This fraction represents the likelihood of the event occurring, expressing it as a ratio of favorable outcomes to the total possible outcomes.
The probability we calculated, 1/45, can also be expressed as a decimal or a percentage for better understanding. To convert the fraction to a decimal, we divide the numerator by the denominator:
1 / 45 ≈ 0.0222
To express this decimal as a percentage, we multiply by 100:
- 0222 * 100 = 2.22%
Therefore, the probability of selecting a two-digit number that is a perfect cube is approximately 0.0222 or 2.22%. This means that if you were to randomly select a two-digit number, there is a 2.22% chance that it would be a perfect cube. This representation of probability provides a clear and intuitive sense of the likelihood of the event occurring.
In conclusion, we have calculated the probability of selecting a two-digit number that is a perfect cube. By systematically identifying the two-digit perfect cubes (27 and 64), determining the total number of two-digit numbers (90), and applying the probability formula, we found the probability to be 1/45, or approximately 2.22%. This exercise demonstrates the application of basic probability principles in a practical mathematical problem. Understanding how to calculate probabilities is essential in various fields, including statistics, finance, and everyday decision-making. By breaking down the problem into manageable steps and applying the appropriate formulas, we can effectively determine the likelihood of specific events occurring.
The process we followed highlights the importance of clear problem definition, systematic calculation, and accurate application of mathematical concepts. Probability calculations provide valuable insights into the likelihood of events, enabling informed decision-making and a deeper understanding of the world around us. This specific example of calculating the probability of selecting a two-digit cube number serves as a fundamental illustration of these principles.
By mastering these techniques, one can apply similar approaches to a wide range of probability problems, enhancing their analytical and problem-solving skills in mathematics and beyond. The ability to calculate and interpret probabilities is a crucial skill in today's data-driven world, where understanding the likelihood of events is essential for making informed choices.