Problem 45
Question
The decimal parts of the logarithms of two numbers taken at random are found to six places. Probability that second can be subtracted first one without borrowing is (A) \(\left(\frac{9}{20}\right)^{6}\) (B) \(\frac{1}{2^{6}}\) (C) \(\left(\frac{11}{20}\right)^{6}\) (D) none of these
Step-by-Step Solution
Verified Answer
The probability is \( \frac{1}{64} \), matching option B.
1Step 1: Understand the Problem
We need to find the probability that the second decimal part can be subtracted from the first decimal part without borrowing digits. This involves calculating probabilities related to randomly selected decimal digits.
2Step 2: Analyze the Decimal Parts
Decimal parts typically range from 0 to 999,999. To focus on individual digits, consider each as a number from 0 to 9, meaning they can be treated with uniform distribution.
3Step 3: Identifying the Criterion for No Borrowing
For no borrowing to be necessary in subtraction, each digit of the second number must be greater than or equal to the corresponding digit of the first number.
4Step 4: Calculate the Probability for a Single Digit
The probability for a single digit of the second number being greater than or equal to the first is 0. Any digit in the first can pair with one equal digit and four greater digits (as 9 choices in decimal). Thus, the probability for a single digit is 0.5.
5Step 5: Extend to Six Digits
As the probability is independent for each of the six decimal places, calculate the total by raising the single digit probability to the power of 6: \( \left(\frac{1}{2}\right)^6 \).
6Step 6: Solve and Identify Correct Option
Evaluating \( \left(\frac{1}{2}\right)^6 \), we get \( \frac{1}{64} \). Hence, the correct option is \( \frac{1}{64} \), matching option B.
Key Concepts
Decimal numbersLogarithmsSubtraction without borrowing
Decimal numbers
When dealing with mathematical problems like the one given in the exercise, understanding decimal numbers is crucial. Decimal numbers are numbers expressed in the base 10 system, which means they consist of digits from 0 to 9 placed in powers of 10. Typically, they represent parts of a whole, especially when we focus on the fractional component of a number.
In the base 10 system, the position of each digit is based on a power of 10. For example, in the number 0.456, the digit '4' is in the tenths place (or \(4 \ imes 0.1\)), '5' is in the hundredths place (or \(5 \ imes 0.01\)), and '6' is in the thousandths place (or \(6 \ imes 0.001\)).
When it comes to probabilities and calculations involving decimal numbers, the properties and arrangement of these digits can influence our solutions. This setup allows for precise arithmetic operations and the opportunity to apply principles such as uniform probability distribution, where each digit has an equal chance of being any number from 0 to 9.
In the base 10 system, the position of each digit is based on a power of 10. For example, in the number 0.456, the digit '4' is in the tenths place (or \(4 \ imes 0.1\)), '5' is in the hundredths place (or \(5 \ imes 0.01\)), and '6' is in the thousandths place (or \(6 \ imes 0.001\)).
When it comes to probabilities and calculations involving decimal numbers, the properties and arrangement of these digits can influence our solutions. This setup allows for precise arithmetic operations and the opportunity to apply principles such as uniform probability distribution, where each digit has an equal chance of being any number from 0 to 9.
Logarithms
Logarithms are an essential mathematical concept that help to solve equations involving exponential relationships. They answer the question: "To what exponent must a base number be raised, to produce a given number?" For instance, if you need to evaluate \( \log_{10}(100) \), you are looking for the power to which 10 must be raised to get 100, which is 2.
When working with logarithms, especially in probabilistic exercises such as the one given, it's often about the mantissa — the decimal part of the logarithm. In real-world applications, these parts can represent rates of growth or scales in many scientific fields. In our exercise, the focus is on evaluating probabilities dealing with these decimal parts.
Using logarithms, you can simplify complex multiplications to addition problems, which are easier to compute. They also provide a way to model dynamical systems that grow exponentially, such as populations or radioactive decay, by turning multiplication and division into easier-to-handle addition and subtraction.
When working with logarithms, especially in probabilistic exercises such as the one given, it's often about the mantissa — the decimal part of the logarithm. In real-world applications, these parts can represent rates of growth or scales in many scientific fields. In our exercise, the focus is on evaluating probabilities dealing with these decimal parts.
Using logarithms, you can simplify complex multiplications to addition problems, which are easier to compute. They also provide a way to model dynamical systems that grow exponentially, such as populations or radioactive decay, by turning multiplication and division into easier-to-handle addition and subtraction.
Subtraction without borrowing
Subtraction without borrowing is a straightforward mathematical operation, particularly in elementary arithmetic. It occurs when every digit of the subtrahend (the number being subtracted) is less than or equal to the corresponding digit of the minuend (the number from which another number is subtracted).
For instance, subtracting 123 from 456: each digit in 123 is smaller than its counterpart in 456, so no borrowing is needed. In tasks that involve longer numbers, such as those seen in scientific calculations or large arithmetic operations, maintaining a subtraction that doesn’t require borrowing simplifies the work and reduces computational complexity.
In the context of the exercise, the ability to subtract without borrowing from decimal parts allows for straightforward calculation of probabilities. This is because each digit in the subtrahend and minuend can be dealt with independently, simplifying the analysis. When each digit of a six-digit number meets this criterion, it ensures the whole subtraction can proceed without needing to adjust any digit downward, which matches well with probabilistic modeling discussed in this type of exercise.
For instance, subtracting 123 from 456: each digit in 123 is smaller than its counterpart in 456, so no borrowing is needed. In tasks that involve longer numbers, such as those seen in scientific calculations or large arithmetic operations, maintaining a subtraction that doesn’t require borrowing simplifies the work and reduces computational complexity.
In the context of the exercise, the ability to subtract without borrowing from decimal parts allows for straightforward calculation of probabilities. This is because each digit in the subtrahend and minuend can be dealt with independently, simplifying the analysis. When each digit of a six-digit number meets this criterion, it ensures the whole subtraction can proceed without needing to adjust any digit downward, which matches well with probabilistic modeling discussed in this type of exercise.
Other exercises in this chapter
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