Chapter 16: Problem 4

Suppose you receive an average of 4 phone calls per day. What is the probability that on a given day you receive no phone calls? Just one call? Exactly 4 calls?

Short Answer

Expert verified

The probability of receiving no calls is \text{e}^{-4}. The probability of receiving one call is 4\text{e}^{-4}. The probability of receiving exactly 4 calls is \frac{32}{3}\text{e}^{-4}.

Step by step solution

Define the scenario

The problem involves determining the probability of receiving a specific number of phone calls in a day when the average number of calls received per day is 4. This is a Poisson distribution problem.

Recall the Poisson probability formula

The probability of observing exactly k events in a Poisson distribution is given by \[ P(X = k) = \frac{\text{e}^{-\text{λ}} \text{λ}^k}{k!} \]Where λ (lambda) is the average rate (4 in this case) and k is the number of events.

Calculate the probability of receiving no phone calls (k=0)

Substitute λ = 4 and k = 0 into the formula:\[ P(X = 0) = \frac{\text{e}^{-4} \times 4^0}{0!} \]This simplifies to:\[ P(X = 0) = \text{e}^{-4} \]

Calculate the probability of receiving exactly one phone call (k=1)

Substitute λ = 4 and k = 1 into the formula:\[ P(X = 1) = \frac{\text{e}^{-4} \times 4^1}{1!} \]This simplifies to:\[ P(X = 1) = 4 \times \text{e}^{-4} \]

Calculate the probability of receiving exactly 4 phone calls (k=4)

Substitute λ = 4 and k = 4 into the formula:\[ P(X = 4) = \frac{\text{e}^{-4} \times 4^4}{4!} \]This simplifies to:\[ P(X = 4) = \frac{256 \times \text{e}^{-4}}{24} \]Simplifying further gives:\[ P(X = 4) = \frac{256}{24} \times \text{e}^{-4} = \frac{32}{3} \times \text{e}^{-4} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena. The central idea revolves around the concept of a 'random variable,' which is a variable that can take on different values depending on the outcome of a random event.

In our context, the random variable represents the number of phone calls received in a day. The probability of each outcome (number of phone calls) is determined using the principles of probability theory.

Important basic concepts of probability theory include:

Probability distribution: Describes the likelihood of different outcomes.
Random events: Events that occur in an unpredictable manner.

In essence, probability theory provides the foundational tools needed to solve problems like the one involving phone calls.

Poisson Probability Formula

The Poisson distribution is used to model the number of times an event occurs in an interval of time or space. Named after the French mathematician Siméon Denis Poisson, it is especially useful when the events occur independently.

The formula for the Poisson probability is expressed as:

\( P(X = k) = \frac{\text{e}^{-\text{λ}} \text{λ}^k}{k!} \)

In this equation:

k: The exact number of events we are interested in (e.g., number of phone calls).
λ: The average event rate over a given time period (average number of phone calls per day).
e: The base of the natural logarithm, approximately equal to 2.71828.

By plugging the values of λ and k into the formula, we can calculate the probability of observing exactly k events.

Event Rate λ

In probability theory and statistics, the event rate, commonly denoted by the Greek letter λ (lambda), represents the average number of times an event occurs within a specific interval. This could be time, area, volume, or any other measure relevant to the problem at hand.

For instance, if λ is 4 in our phone call problem, it means that, on average, 4 phone calls are received per day.

The significance of λ in the Poisson distribution is crucial because it directly influences the probabilities of different counts of the event occurring. To reiterate:

When λ is higher, the probabilities of higher counts (k) occurring increase.
When λ is lower, the event of interest occurs less frequently, resulting in lower probabilities for higher counts (k).

Hence, understanding and accurately determining λ is essential for making predictions using the Poisson distribution.