Question 1201238
**1. Determine the Expected Number of Diners at Feliks's Cafe During Dinner**

* **Total number of dinners:** 
    * n * P2% = 2105 * (52/100) = 1094.6 dinners

* **Expected number of diners at each cafe during dinner:**
    * 1094.6 dinners / m cafes = 1094.6 / 7 ≈ 156.37 diners/cafe

**2. Determine the Required Number of Seats to Avoid Overfilling r% of Evenings**

* **Calculate the standard deviation:** 
    * Assuming independent choices of cafes by vacationers, the number of diners at a cafe during dinner can be approximated by a binomial distribution.
    * For a binomial distribution, the standard deviation is: 
        * σ = √(n * p * (1 - p)) 
        * where n is the total number of dinners (1094.6), and p is the probability of a diner choosing a specific cafe (1/m = 1/7) 
        * σ = √(1094.6 * (1/7) * (6/7)) ≈ 11.84 diners

* **Determine the z-score corresponding to the desired overfilling percentage (r%):**
    * Find the z-score such that P(Z > z) = r/100, where Z is a standard normal random variable. 
    * For r = 8%, use a standard normal distribution table or calculator to find the corresponding z-score. Let's assume you find z = 1.41 (approximately).

* **Calculate the number of seats needed (x):**
    * x = Expected number of diners + (z * standard deviation)
    * x = 156.37 + (1.41 * 11.84) ≈ 175.12

* **Round up to the nearest integer:** 
    * x = 176 seats

**Therefore, Feliks's cafe should have at least 176 seats to avoid being overfilled on more than 8% of evenings during dinner.**

**3. Probability of Overfilling During Breakfast**

* **Calculate the expected number of diners during breakfast:**
    * n * P1% = 2105 * (58/100) = 1220.9 breakfasts

* **Expected number of diners at each cafe during breakfast:**
    * 1220.9 breakfasts / m cafes = 1220.9 / 7 ≈ 174.41 breakfasts/cafe

* **Calculate the standard deviation:** 
    * σ = √(n * p * (1 - p)) 
    * σ = √(1220.9 * (1/7) * (6/7)) ≈ 12.57 breakfasts

* **Determine the z-score for the current number of seats (assuming 176 seats):**
    * z = (176 - 174.41) / 12.57 ≈ 0.126

* **Find the probability of exceeding the number of seats:**
    * P(Overfilling) = P(Z > 0.126) 
    * Use a standard normal distribution table or calculator to find this probability.

**Therefore, the probability that Feliks's cafe will be overfilled during breakfast with 176 seats can be calculated using the z-score and the standard normal distribution table.**

**Important Notes:**

* This analysis assumes that the choices of cafes by different vacationers are independent.
* The actual probability of overfilling may vary slightly due to the approximation using the normal distribution (Moivre-Laplace theorem).
* This calculation provides an estimate. Actual results may differ based on various factors such as seasonality, special events, and unforeseen circumstances.

I hope this helps!