Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Answer

The probability is \(\frac{1}{2^6}\).

Step 1: Understanding the Problem

We have two random numbers and their logarithms. The problem asks for the probability that the decimal part of the second logarithm can be subtracted from the first one without borrowing.

Step 2: Understanding Decimal Parts

Each decimal part is essentially a number between 0 and 1. We are working with these decimal parts to find the probability condition mentioned.

Step 3: Condition for No Borrowing

To subtract two decimal parts without borrowing, the first decimal part must be greater than or equal to the second decimal part.

Step 4: Probabilistic Model

Since each decimal part can be any number between 0 and 1, the probability that one is greater than the other is equivalent to comparing two random numbers within this range.

Step 5: Computing the Probability

Both decimal parts are uniformly distributed over the interval \([0, 1)\). The condition for not borrowing translates to: for two numbers \(x\) and \(y\), the probability that \(x - y \geq 0\) is \(P(x \geq y)\). This probability is \(\frac{1}{2}\) for each digit position.

Step 6: Calculating for Six Decimal Places

We are looking at six decimal places. Since each decimal position must satisfy the no borrowing condition independently, the probability for all six places is \(\left(\frac{1}{2}\right)^6\).

Step 7: Solution

The final probability, considering all decimal places, is \(\frac{1}{2^6}\). Thus, the answer is \( \frac{1}{2^{6}} \).

Key Concepts

Logarithms

Logarithms are a foundational concept in mathematics, used to determine the power to which a number must be raised to obtain another number. Simply put, if we have \[ b^x = a \]then \[ x = \log_b{a} \]. This function helps in solving equations involving exponentials by converting them into simpler operations.
Logarithms are often expressed in terms of a decimal system. For example, common logarithms use base 10, and natural logarithms use base \( e \), where \( e \approx 2.718 \). This makes them extremely useful in various fields like sciences, engineering, and even financial modeling.
In this exercise, we consider the decimal part of logarithms, which essentially means we are interested in the fractional components after the integer part. Understanding these parts is crucial when dealing with probabilities, as they are uniformly distributed between 0 and 1. This uniform distribution plays a significant role in calculating the probability conditions like in the given example.

Decimal places in probability

When dealing with the probability of events, understanding decimal places is crucial, especially in precise calculations. In probability, decimal places often describe the accuracy to which a probability is expressed. For instance, a probability such as 0.5 refers to a 50% chance of an event occurring.
In the context of this exercise, we are examining the decimal parts of logarithmic values. These parts are used to determine probabilities concerning the subtraction without borrowing.
Each decimal digit is independently crucial, as it contributes to the overall probability output when multiplied across several decimal places. Setting an individual probability value of \(\frac{1}{2}\) for each independent event, as in our problem concerning subtracting without borrowing, results in calculating across multiple positions by raising to the power equivalent to the number of decimal places – in this case, 6.
This meticulous computation ensures that the final probability is accurately presented, as seen in the step-by-step solutions.

Uniform distribution

Uniform distribution is a type of probability distribution in which all outcomes are equally likely. Consider a simple example of flipping a fair coin, where heads and tails each have an equal probability of occurrence, \(\frac{1}{2}\) each.
In this exercise, we analyze the decimal parts of two logarithmic numbers, which are modeled as being uniformly distributed over the interval \([0, 1)\). This means that any number within this range is equally probable to occur.
Uniform distribution becomes particularly useful when you need to calculate probabilities involving any two numbers within a specific range. For example, determining whether a randomly selected decimal is greater than another, which is directly applied in the condition of subtracting without borrowing where \(P(x \geq y)\) translates to \(\frac{1}{2}\).
Grasping the uniform distribution's fundamentals aids in solving this exercise effectively, as it provides a clear understanding of the randomness associated with decimal parts and the probability calculations that follow.