Question

Find the probabilities for \(x\) using the Poisson formula. $$ \mu=3 ; P(x=0), P(x=1), \text { and } P(x>1) $$

Short answer

Question: On an average, 3 calls per hour are received at a help desk. Calculate the probabilities of receiving 0, 1, and more than 1 call using the Poisson distribution. Answer: The probabilities are as follows: - Probability of receiving 0 calls: \(P(x=0) \approx 0.0498\) - Probability of receiving 1 call: \(P(x=1) \approx 0.1494\) - Probability of receiving more than 1 call: \(P(x>1) \approx 0.8008\)

Step-by-step solution

Find \(P(x=0)\) using the Poisson formula

Apply the Poisson formula for \(x=0\): $$ P(x=0) = \frac{e^{-3} \cdot 3^0}{0!} $$

Evaluate \(P(x=0)\)

Calculate the values: $$ P(x=0) = \frac{e^{-3} \cdot 1}{1} = e^{-3} \approx 0.0498 $$

Find \(P(x=1)\) using the Poisson formula

Apply the Poisson formula for \(x=1\): $$ P(x=1) = \frac{e^{-3} \cdot 3^1}{1!} $$

Evaluate \(P(x=1)\)

Calculate the values: $$ P(x=1) = \frac{e^{-3} \cdot 3}{1} = 3e^{-3} \approx 0.1494 $$

Find \(P(x>1)\) using the complement rule

Since we know \(P(x=0)\) and \(P(x=1)\), we can find \(P(x>1)\) by using the complement rule: $$ P(x>1) = 1 - P(x=0) - P(x=1) $$

Evaluate \(P(x>1)\)

Calculate the final probability: $$ P(x>1) = 1 - 0.0498 - 0.1494 \approx 0.8008 $$ The probabilities for the three cases are approximately: $$ P(x=0) \approx 0.0498 \\ P(x=1) \approx 0.1494 \\ P(x>1) \approx 0.8008 $$

Key concepts

Understanding Probability

Probability is a fundamental concept in statistics, dealing with the measurement of how likely an event is to occur. It's a value between 0 and 1, inclusive, where 0 indicates the event will not occur, and 1 indicates certainty of the event's occurrence. In our daily lives, we often estimate the chances of events happening, such as it raining today or winning a game.

When dealing with probabilities, various methods are used to calculate or predict the chances of specific events. This includes simple probabilities for coin tosses or more complex models like the Poisson distribution, which helps assess the probability of a given number of events happening within a fixed interval of time or space.

In the provided exercise, we are faced with calculating specific probabilities for the number of occurrences within a defined scope—these calculations hinge on understanding and accurately applying the Poisson formula.

Applying the Poisson Formula

The Poisson formula is a mathematical expression used within the framework of the Poisson distribution to predict the probability of a certain number of events occurring over a fixed period, given the average number of times the event occurs over that period. The formula is given by:
\[ P(x) = \frac{e^{-\mu} \cdot \mu^x}{x!} \]
where \( P(x) \) denotes the probability of \( x \) events occurring, \( \mu \) is the average number of occurrences, \( e \) is Euler's number (approximately 2.71828), and \( x! \) is the factorial of \( x \). This formula is notably powerful when dealing with events that occur independently and at a constant rate.

In the original exercise, the formula is applied to find probabilities for different values of \( x \), with \( \mu \) equal to 3. The calculation of these probabilities is the core exercise, demanding a clear understanding of how to manipulate the formula for different values of \( x \).

Leveraging the Complement Rule

The complement rule is another essential principle in probability theory. It is a straightforward approach to determine the likelihood of an event not happening by subtracting the probability of the event occurring from 1. The rule stems from the axiom that the sum of the probabilities of all possible outcomes in a probability space must equal 1. Hence, for any event \( A \), the probability that \( A \) does not occur, denoted as \( P(eg A) \), is given by:
\[ P(eg A) = 1 - P(A) \]
When combined with the Poisson formula, the complement rule can simplify the process of finding probabilities for multiple events. In the context of the homework problem, the rule assists in determining \( P(x>1) \) by subtracting the probabilities of \( x=0 \) and \( x=1 \) from 1, which is a practical application of how these concepts interrelate to provide comprehensive solutions to probabilistic scenarios.

Each concept — probability, the Poisson formula, and the complement rule — plays a pivotal role in accurately solving problems related to event occurrences. Gaining proficiency in these areas opens the door to understanding and predicting outcomes in various real-life and academic scenarios.