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

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 probability is \(\frac{k}{1,000}\) which matches option (B).

Step 1: Understanding the Problem

We need to determine the probability that a locker, which opens with a specific three-digit code, is opened by a stranger who guesses the code at random after up to \(k\) trials, where \(k < 1,000\).

Step 2: Calculating Total Outcomes

The locker code is a three-digit number. Therefore, there are 1,000 possible codes since each digit can be any number from 0 to 9. Thus, the total number of possible three-digit codes is given by \(10 \times 10 \times 10 = 1,000\).

Step 3: Finding Successful Outcomes

For each trial, there is only one successful outcome where the guessed code matches the chosen code, making it equal to one successful outcome per trial. Therefore, in \(k\) trials, there are \(k\) successful outcomes.

Step 4: Calculating the Probability

The probability of the locker being opened in \(k\) trials is given by the ratio of successful outcomes to total outcomes. Hence, the probability is \(\frac{k}{1,000}\).

Step 5: Selecting Correct Option

After determining that the probability is \(\frac{k}{1,000}\), we compare this with the given options. This matches option (B), \(\frac{k}{1000}\).

Key Concepts

Three-digit code

A three-digit code is a sequence of numbers used to secure access to various things, such as lockers and safes. Each digit in a three-digit code can range from 0 to 9. Therefore, you might imagine combinations like 000, 123, or 999.
This results in a total of 1,000 possible combinations, calculated as follows:

• Each of the three positions in the code can hold one of 10 different digits (from 0 to 9).
• Thus, for the first digit, you have 10 choices; for the second digit, another 10 choices; and for the third as well, another 10 choices.
• This results in a total of 10 x 10 x 10 = 1,000 possible codes.

Understanding the concept of a three-digit code is crucial to solving problems involving code guessing or securing items.

Combinatorics

Combinatorics is a branch of mathematics primarily concerned with counting, arranging, and finding patterns in sets. When dealing with scenarios involving random guessing, such as the locker problem, understanding combinatorics is essential.
Consider the process of guessing the three-digit code:

• The locker involves selecting one correct combination out of 1,000 possibilities.
• Within the field of combinatorics, this kind of problem is tackled by calculating the total number of possible combinations or arrangements and identifying successful outcomes.

The basic idea is to systematically count the number of possibilities (total outcomes) and how many of those meet a particular condition (successful outcomes). Here, combinatorics helps us understand that the correct code is just one of many possible arrangements, reinforcing the low probability of random success in guessing a code correctly.

Probability calculation

Probability is the measure of how likely an event is to occur. It is calculated as the ratio of successful outcomes to total possible outcomes.
In our locker scenario:

• The total number of possible outcomes is 1,000, as calculated by the total number of three-digit codes.
• For each guess, there is exactly one successful outcome – the exact code chosen by the passenger. Thus, in one attempt, the chance of opening the locker is \(\frac{1}{1,000}\).

However, if a stranger tries multiple attempts, the probability changes. If the stranger makes \(k\) random attempts, the probability that one of these is successful is given by \(\frac{k}{1,000}\).
This formula tells us that the more attempts allowed (up to 999), the higher the chance to correctly guess the code, although this chance remains quite small. Calculating probability in this way is a core technique for evaluating risk and uncertainty in different scenarios.