Problem 89 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

At a railway station a passenger leaves his luggage in a locker which is opened by dialling a three-digit code (say, \(253,009,325 \ldots\) ). 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, \(k<1,000\) ) (A) \(\frac{k}{100}\) (B) \(\frac{k}{1000}\) (C) \(\frac{1,000-k}{1,000}\) (D) none of these

Step-by-Step Solution

Answer

The correct option is (D) none of these.

Step 1: Understanding the Problem

We need to find the probability that a three-digit locker code matches and opens the locker after exactly \(k\) random attempts by the stranger. There are 1,000 possible three-digit combinations (000 to 999).

Step 2: Defining the Total Outcomes

The total number of possible outcomes is the total number of possible codes, i.e., 1,000. This is because there are digits ranging from 000 to 999, giving us 1,000 combinations.

Step 3: Calculating Successful Outcomes

For the locker to be opened after exactly \(k\) trials, the stranger must fail \(k-1\) times and succeed on the \(k^{th}\) trial. This is similar to getting exactly one success in \(k\) trials on the \(k^{th}\) try, with the success probability being \(\frac{1}{1000}\) once and the failure for the \(k-1\) times with probability \(\frac{999}{1000}\).

Step 4: Using the Geometric Distribution

The situation follows a geometric distribution where the probability of succeeding on exactly the \(k^{th}\) trial is given by \[(\text{failure probability})^{\text{number of failures}} \times \text{success probability} = \left(\frac{999}{1000}\right)^{k-1} \times \frac{1}{1000}.\]

Step 5: Comparing with Options

The given options do not mathematically match the probability we calculated using the geometric distribution. Therefore, based on this understanding, the correct option for the problem must be (D) none of these.

Key Concepts

Geometric Distribution

The geometric distribution is a probability distribution that models the number of trials needed for the first success in repeated Bernoulli trials, each with the same probability of success. In this scenario, a trial represents the attempt of cracking the locker code. The success here is when the locker opens with the correct code.

Understanding this can be quite straightforward when you recognize the key characteristics:

• Each trial is independent, meaning the outcome of one does not affect another.
• The probability of success remains constant in each trial (here, the probability of choosing the correct code \( \frac{1}{1000} \)).
• We are interested in the number of trials up to and including the first success, which in mathematical terms can be expressed as \( (1-p)^{k-1} \times p \), where \( p \) is the success probability, and \( k \) is the specific trial in which success occurs.

When using this distribution, you are often asked about the probability of success on a specific trial, which can be directly calculated using the formula mentioned.

Probability of Success

Probability of Success is a fundamental concept in probability theory. In the context of the exercise, it defines the chance of successfully opening the locker with the specific code set by the passenger. Because there are 1,000 different possible combinations (ranging from 000 to 999), this probability is quite small.

Let's break it down:

• The probability that on any single attempt, the stranger successfully guesses the correct code is \( \frac{1}{1000} \). Thus, the stranger has just one success out of the 1,000 possible codes.
• If the stranger makes multiple attempts, every single attempt still has a \( \frac{1}{1000} \) chance of success without affecting previous or future trials since each try is independent.

Having a uniform probability across each attempt means that the likelihood remains consistent no matter how many failed attempts were made previously.

Three-digit Combinations

Three-digit combinations are utilized here to securely lock the luggage. This concept relies on the permutation of digits in three positions to create a range of unique codes. Each place in these three digits can be filled by any digit from 0 to 9.

Here’s how it breaks down:

• The first digit can be anything from 0 to 9, giving us 10 possibilities.
• The second digit also has 10 possible values (0-9).
• And similarly, the third digit has another 10 options available.

By multiplying these possibilities (10 for each digit), we compute the total number of unique combinations: \( 10 \times 10 \times 10 = 1000 \) different three-digit codes.

This spread of combinations ensures that guessing the correct code by random dialing is quite challenging, highlighting the need for understanding probability theory to decipher such a task.

Random Trial Analysis

Random Trial Analysis encompasses the study and comprehension of events that happen during random trials, such as the attempts to unlock the locker. Each attempt by the stranger is a trial, and the analysis revolves around predicting the likelihood of events based on those trials.

When analyzing such trials, consider these points:

• Each dial attempt is a trial that can either result in success (finding the correct code) or failure (wrong code).
• The stranger makes multiple independent attempts where past results do not impact future ones.
• Outcome analysis focuses on both specific trial outcomes (probability of success on exactly the \( k^{th} \) trial) and cumulative outcomes over several trials.

This method of analysis reinforces the concepts of independence and probability, providing a structured way to predict outcomes and support decision-making processes about expected probabilities after multiple attempts.