Question 1201238: In a resort town during the season, there are n vacationers. Each vacationer has breakfast approximately P1% of the time and dinner P2% of the time at a café. Each time a vacationer chooses a café randomly. There are m cafés in the town. Entrepreneur Feliks wants to set up another one. What is the minimum number of seats needed in this café so that it is not overfilled more than r% of evenings during dinner? What is the probability that such a café will be overfilled during breakfast? Use the Moivre-Laplace theorem. ([n, m, P1, P2, r] = [2105, 7, 58, 52, 8])
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! **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!
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