Problem 63 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

At a telephone enquiry system the number of phone calls regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is (a) \(\frac{6}{5^{\mathrm{e}}}\) (b) \(\frac{5}{6}\) (c) \(\frac{6}{55}\) (d) \(\frac{6}{\mathrm{e}^{5}}\)

Step-by-Step Solution

Answer

The probability is \(\frac{6}{e^{5}}\), which corresponds to option (d).

Step 1: Understand the Poisson Distribution

A Poisson distribution describes the probability of a number of events occurring in a fixed interval of time or space. Here, the average rate (\(\lambda\)) is 5 calls per 10 minutes.

Step 2: Define the Probability of At Most One Call

"At most one call" means calculating the probabilities of getting either 0 or 1 call and adding them together: \(P(X \leq 1) = P(X = 0) + P(X = 1)\).

Step 3: Calculate the Probability of Zero Calls

Use the Poisson probability formula: \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\). Substitute \(k = 0\): \[P(X = 0) = \frac{5^0 e^{-5}}{0!} = \frac{1}{e^{5}}\].

Step 4: Calculate the Probability of One Call

Again, use the Poisson formula for \(k = 1\): \[P(X = 1) = \frac{5^1 e^{-5}}{1!} = \frac{5}{e^{5}}\].

Step 5: Add the Probabilities for At Most One Call

Add the probabilities calculated in Steps 3 and 4: \[P(X \leq 1) = \frac{1}{e^{5}} + \frac{5}{e^{5}} = \frac{6}{e^{5}}\].

Step 6: Identify the Correct Option

Compare \(\frac{6}{e^{5}}\) with the given options. The correct answer is option (d) \(\frac{6}{e^{5}}\).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides a framework for quantifying uncertainty and making predictions about future events. Here, in the context of a telephone enquiry system, probability theory is employed to predict the number of phone calls received within a specific time frame. By defining the probability of events, like receiving a specific number of calls, this concept helps manage various components of uncertainty in everyday life.
When dealing with probabilities, some essential elements are:

• Events: The outcomes or occurrences we study, like receiving calls.
• Sample space: All possible events in a scenario.
• Probability: A numerical value, between 0 and 1, representing the likelihood of an event.

In this exercise, our focus is on predicting call frequency using a specific type of probability distribution, which will help us determine the probability of receiving a certain number of calls in a given time period.

Statistical Distribution

A statistical distribution describes how the probability of an event varies across different values. It's an organized way of displaying odds. In this case, we're dealing with the Poisson distribution, which is a specific type of statistical distribution.
The Poisson distribution is useful for scenarios where we're counting occurrences of events over an interval, like phone calls in a given time frame. It requires us to know:

• The average rate (\(\lambda\)): the expected number of occurrences per interval. Here, \(\lambda = 5\) calls every 10 minutes.
• The specific intervals: i.e., the time frame for counting, in this case, a 10-minute period.

The Poisson formula is \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\), where :

• \(k\) is the number of events we want to calculate the probability for.
• \(e\) is Euler's number, an important constant in mathematics.

Utilizing this formula helps determine the likelihood of receiving any number of calls within the specified period.

Telephone Enquiry System

A telephone enquiry system is a service that answers incoming calls regarding specific questions or issues by customers. Such systems are crucial for customer support and service operations. Understanding the probability of call volume helps improve efficiency, resource management, and customer satisfaction.
By using a Poisson distribution, organizations can anticipate call volumes. They can strategically plan staffing, so customer queries are addressed swiftly. Here's how accurate call predictions benefit the system:

• Ensures enough staff to manage typical call volumes, preventing overloading.
• Optimizes scheduling, balancing workforce and expected inquiries.
• Improves customer experience by reducing wait times and enhancing service quality.
• Assists in decision-making regarding system upgrades or changes.

Adopting statistical and probability tools offers a competitive advantage, as it aligns service capacity with customer demand effectively.