Problem 44 - A Complete Resource Book in Mathematics for JEE Main

Question

At a railway staition a passenger leaves his luggage in a locker which is opened by dialling a three digit code (say \(253,009,325\) etc.). The passenger chooses the code, closes the locker and leaves for the town. A strange man, who does not know the code, tries to open the locker by dialling three digits at random. The probability that the locker opens after \(k\) trials is (Here \(k<1000\) ) (A) \(\frac{k}{100}\) (B) \(\frac{k}{1000}\) (C) \(\frac{1000 \quad k}{1000}\) (D) none of these.

Step-by-Step Solution

Answer

The probability that the locker opens after \(k\) trials is (B) \( \frac{k}{1000} \).

Step 1: Understanding the Problem

The locker opens with a 3-digit code, meaning there are 1,000 possible combinations (000 to 999). A stranger trying random codes has a certain probability of opening the locker with a specific trial.

Step 2: Calculate Probability of Success on One Trial

The probability of opening the locker on any random trial is the number of successful outcomes (correct code) divided by the total possible outcomes (all codes). This is \( \frac{1}{1000} \).

Step 3: Probability of Success After 'k' Trials

For the stranger to succeed within the first \(k\) trials, at least one of those \(k\) attempts must be successful. The probability of success on each trial is \( \frac{1}{1000} \). Therefore, after \(k\) trials, the probability is \( \frac{k}{1000} \).

Step 4: Choose the Correct Option

After finding that the probability after \(k\) trials is \( \frac{k}{1000} \), we match this expression with one of the given options. The correct answer is (B) \( \frac{k}{1000} \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns among objects. In this exercise about a three-digit code lock, combinatorics helps us determine how many possible combinations exist. For a three-digit code where each digit can be anything from 0 to 9, we calculate the total possible combinations by multiplying the number of choices for each digit.

• First digit: 10 choices (from 0 to 9)
• Second digit: 10 choices (from 0 to 9)
• Third digit: 10 choices (from 0 to 9)

By multiplying these together, we get the total number of combinations: \(10 \times 10 \times 10 = 1000\).Combinatorics provides a structured way to think about how these combinations are formed and counted. Without it, solving problems involving multiple configuration possibilities could be cumbersome.

Three-digit code

A three-digit code consists of three numbers, each ranging from 0 to 9. These codes are commonly used in locks and security systems, such as the locker in the train station mentioned in the exercise. To understand why there are 1000 possible codes, consider that each digit in the code can be any integer between 0 and 9.

• This gives us 10 possibilities for each digit independently.
• Since these choices are independent, we multiply them together to get the total number of combinations.
• Therefore, the calculation is: \(10 \times 10 \times 10 = 1000\).

This multiplication represents the basic principle of counting in combinatorics. By understanding and utilizing these principles, you can quickly estimate the total configurations for any similar scenario involving multiple-choice options spread across multiple slots.

Successive trials probability

Probability in successive trials involves understanding the chance of an event occurring at least once after a series of attempts. In this scenario, a stranger tries to open a locker by randomly selecting codes. Let's explore the process of finding the probability of success after several trials.

• Each trial has a probability of success equal to \( \frac{1}{1000} \), as there is only one correct code among 1000 possibilities.
• If a span of \(k\) trials is considered, the probability that at least one attempt is successful is the cumulative probability across all trials.
• This cumulative probability can be estimated as \( \frac{k}{1000} \) when considering up to \(k\), where the number of trials is less than 1000.

The concept of successive trials probability is essential for calculating the likelihood of achieving a successful outcome over a series of independent attempts. It's crucial to distinguish between this aggregated probability and the probability of each individual trial. This helps in crafting strategies where multiple attempts are possible and makes it easier to anticipate the success rate over time.