Problem 159 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

At a telephone enquiry system the number of phone cells 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 [2006] (A) \(\frac{6}{5^{c}}\) (B) \(\frac{5}{6}\) (C) \(\frac{6}{55}\) (D) \(\frac{6}{e^{5}}\)

Step-by-Step Solution

Answer

The correct answer is (D) \( \frac{6}{e^5} \).

Step 1: Identify the Parameters

We are given that the number of phone calls follows a Poisson distribution with an average, \( \lambda \), of 5 calls in a 10-minute interval.

Step 2: Poisson Probability Formula

The formula for the Poisson probability of observing \( k \) events is: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where \( \lambda = 5 \).

Step 3: Calculate Probability for No Calls

Substitute \( k = 0 \) into the Poisson formula: \[ P(X = 0) = \frac{e^{-5} \cdot 5^0}{0!} = e^{-5} \]

Step 4: Calculate Probability for One Call

Substitute \( k = 1 \) into the Poisson formula: \[ P(X = 1) = \frac{e^{-5} \cdot 5^1}{1!} = \frac{5e^{-5}}{1} = 5e^{-5} \]

Step 5: Sum Probabilities for At Most One Call

At most one call means 0 or 1 call: \[ P(X \leq 1) = P(X = 0) + P(X = 1) = e^{-5} + 5e^{-5} = 6e^{-5} \]

Step 6: Select Correct Option

The probability that there is at most one phone call is \[ 6e^{-5} \]. Comparing this with the options provided, the correct choice is (D) \( \frac{6}{e^5} \).

Key Concepts

probability theory

Probability theory is a branch of mathematics that deals with calculating the likelihood of various outcomes in uncertain situations. In simple terms, it helps us understand how probable an event is to occur.
In probability theory, events are analyzed using concepts like random variables and probability distributions. A random variable is a variable whose values depend on outcomes of a random phenomenon. Probability distributions, on the other hand, describe how these probabilities are spread over different possible outcomes.

The Poisson distribution is one such probability distribution. It helps model the probability of a given number of events happening in a fixed interval of time or space, as long as these events occur with a known constant mean rate and independently of the time since the last event.
It's widely used in fields like telecommunications, genetics, and even physics, where it’s necessary to model events that occur randomly and independently over time. Understanding probability theory provides the foundation needed to apply formulas like the Poisson distribution and to make informed predictions based on data and statistical analysis.

Poisson probability formula

The Poisson probability formula is a crucial tool for calculating the likelihood of a number of events occurring within a specified period. The formula can be expressed as:

• \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \)

Here, \( X \) is a random variable representing the number of events, \( k \) is the actual number of events we want to find the probability for, \( \lambda \) is the average number of events, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

This formula assumes that events occur independently and the rate is constant. When using the Poisson formula, you:

• Identify \( \lambda \), the mean number of events within the time frame.
• Substitute \( k \) with the number of events you're interested in.
• Calculate using basic arithmetic with exponents and factorials.

In our example, the mean rate is 5 calls in a 10-minute interval. To find the probability of no more than one phone call, the formula is used to calculate for \( k = 0 \) and \( k = 1 \), then these probabilities are summed to give the probability of getting 0 or 1 phone call.

mathematical expectation

Mathematical expectation, also known as expected value, is a fundamental concept in probability theory. It represents the average outcome you would expect if you were to repeat an experiment or a process many times.
The expected value is calculated by multiplying each outcome by its probability and summing up all the resulting values. It gives us a kind of 'center' or 'mean' of the distribution of outcomes.

For a Poisson distribution, the mathematical expectation can be directly represented by \( \lambda \). That's because the mean rate of events occurring per time interval in a Poisson process, \( \lambda \), is already the expected number of events.
This simplicity is what makes Poisson distribution convenient in real-life applications, as it avoids complicated calculations to find the average outcome.
The expected value provides insights into what is "normal" or "typical" in the process being studied, allowing you to make informed decisions and predictions based on the average behavior of the system.