Based on past data, the number of complaints received from hotel guests follow a Poisson Distribution with an average of 2 complaints per day. A day with more than 4 complaints is considered a high-complaint day. If the probability that there are more than 1 high-complaint days in a week exceeds 0.05, it would warrant management’s attention. With appropriate calculations, determine if the complaint situation in the hotel warrants management’s attention. give all steps

We are tasked with determining whether the complaint situation at a hotel warrants management's attention based on the probability of having more than one high-complaint day in a week. The number of complaints follows a Poisson distribution with an average of 2 complaints per day, and a day is considered high-complaint if there are more than 4 complaints.

### Step 1: Define Variables and Distributions
- The number of complaints per day, \( X \), follows a Poisson distribution with a mean of 2:
  \[
  X \sim \text{Poisson}(2)
  \]

### Step 2: Probability of a High-Complaint Day
- A high-complaint day occurs when there are more than 4 complaints. We need to find \( P(X > 4) \).

The probability mass function (PMF) for the Poisson distribution is:
\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]
where \( \lambda = 2 \) and \( k \) is the number of complaints.

To calculate \( P(X > 4) \), we first calculate \( P(X \leq 4) \) using the complement rule:
\[
P(X > 4) = 1 - P(X \leq 4)
\]
\[
P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
\]

Let's calculate this probability.

### Step 3: Calculating \( P(X \leq 4) \)

We calculate the individual probabilities using the Poisson PMF:
- \( P(X = 0) = \frac{2^0 e^{-2}}{0!} = e^{-2} \)
- \( P(X = 1) = \frac{2^1 e^{-2}}{1!} = 2e^{-2} \)
- \( P(X = 2) = \frac{2^2 e^{-2}}{2!} = 2^2 \cdot e^{-2}/2 \)
- \( P(X = 3) = \frac{2^3e^{-2}}

Explore how to assess hotel complaint patterns using Poisson distribution with an average of 2 complaints per day. Determine if the probability of more than one high-complaint day in a week warrants management’s attention. Includes step-by-step calculations and probabilities.

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