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The Poisson distribution is a powerful tool used for calculating the probability of a given number of events happening in a fixed interval of time or space. In the context of the problem, we are interested in finding the probability of receiving exactly 3 pieces of junk mail in a day.To calculate this probability, we utilize the Poisson formula, which is:\[ P(x; \mu) = \frac{e^{-\mu} \cdot \mu^x}{x!} \]This formula essentially combines a few concepts including exponential decay, raising a number to a power, and the factorial. We break down each step clearly to ensure all parts of the calculation are understood.In this case,

• \( \mu \) is the average number of pieces of junk mail, which is 1.7.
• \( x \) is the number of pieces we are trying to calculate the probability for, which is 3.
• \( e \) is the mathematical constant approximately equal to 2.71828. It is a crucial part of the formula, representing the natural exponential function.

Using these parameters, we systematically calculate each part of the formula to reach our probability value.

Factorials are a fundamental concept in probability and frequently appear in various probability distributions, including the Poisson distribution. The factorial, denoted with an exclamation point, like \(x!\), refers to the product of an integer and all the positive integers less than it.For example, the factorial of 3, written as \(3!\), is:\[ 3! = 3 \times 2 \times 1 = 6 \]This concept is central in many probability calculations as it helps determine the number of ways events can occur. This occurs because factorials count the permutations or arrangements of events, which are foundational elements in understanding probabilities.In the Poisson context, the factorial is used in the denominator of the formula, normalizing the probability by accounting for the number of ways the specified event could happen.

A probability distribution formula is used to describe the likelihood of different outcomes in an experiment. The Poisson distribution is specifically suited for scenarios where we count the number of times an event happens within a given interval, especially when events occur independently and the mean rate is constant.The formula for the Poisson distribution is:\[ P(x; \mu) = \frac{e^{-\mu} \cdot \mu^x}{x!} \]Key components of this formula include:

• **Probability Expression**: \( P(x; \mu) \) represents the probability of observing exactly \( x \) events given the average rate \( \mu \).
• **Exponential Component**: \( e^{-\mu} \) accounts for the decreasing probability of extremely low or high occurrences around the mean, capturing the natural decay in likelihood.
• **Mean Raised to Power**: \( \mu^x \) represents how likely the precise number of events is, relative to the mean.
• **Normalization by Factorial**: \( x! \) to reflect the actual arrangement count for x events, ensuring results align with probability laws.

This formula provides a way to quantify randomness and predict the probability of certain outcomes in real-life, making it a versatile and essential component in statistics and probability theory.