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

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: Understanding the Problem

The problem involves a Poisson distribution where the average number of events (phone calls) in a given interval is 5. We need to find the probability of receiving at most one phone call in 10 minutes.

Step 2: Poisson Probability Formula

The probability of observing exactly k events in a Poisson distribution is given by the formula: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]where \( \lambda \) is the average number of events (5 in this case).

Step 3: Calculate Probability for 0 Calls

Apply the Poisson probability formula for \( k = 0 \):\[ P(X = 0) = \frac{5^0 e^{-5}}{0!} = e^{-5} \]

Step 4: Calculate Probability for 1 Call

Apply the Poisson probability formula for \( k = 1 \):\[ P(X = 1) = \frac{5^1 e^{-5}}{1!} = 5e^{-5} \]

Step 5: Calculate Probability for at Most 1 Call

The probability of at most one call is the sum of the probabilities of zero and one call:\[ P(X \leq 1) = P(X = 0) + P(X = 1) = e^{-5} + 5e^{-5} = 6e^{-5} \]

Step 6: Choose the Correct Answer

The probability expression \( 6e^{-5} \) matches the option (D) \( \frac{6}{e^5} \).

Key Concepts

Probability

Probability is a concept that helps us understand the likelihood of different outcomes in various scenarios. In the context of the Poisson distribution for phone calls, probability helps us calculate how likely it is to receive a certain number of calls in a time frame.
In this exercise, we're dealing with the probability of receiving either zero or one phone call. Probabilities range from 0 to 1, where 0 means an event is impossible and 1 means it is certain.

• The average number of phone calls: 5 per 10 minutes.
• The task: Find the probability of having at most 1 call, meaning either 0 or 1 call.

The Poisson distribution is specially designed for calculating probabilities of events that occur independently and with a constant mean over a period. This is how it can help determine the probability of several phone calls during an interval.

Mathematical Problem Solving

Mathematical problem solving is like unraveling a mystery, using logic and formulas to find the answer. Here, we deal with a Poisson distribution problem to solve it step-by-step.
To tackle this, you need to follow a structured approach:

• First, identify what is being asked. Here, it's the probability of having at most one call.
• Use the right formula. For Poisson, we use \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]
• Calculate for specific values (0 and 1 calls) using this formula.

For zero phone calls, the calculation results in \( e^{-5} \) since the formula simplifies beautifully when k is 0. For one phone call, the same formula gives us \( 5e^{-5} \). Then, sum these probabilities for the final solution.

Step-by-step Mathematical Solutions

Breaking down problems into manageable steps is essential in math. Step-by-step solutions help in understanding each phase of a problem.
Let's see how this works practically:

• Step 1: Understand the given distribution and requirements.
• Step 2: Apply the Poisson formula for each potential outcome (0 and 1 call).
• Step 3: Sum up the individual probabilities to solve for 'at most one call'.

Finally, match the calculated result \( 6e^{-5} \) with the answer choices given, confirming (D) as the answer. By following these steps, the problem becomes less intimidating and more clear. Each component builds upon the last, leading to a comprehensive understanding and correct solution.