Question 1192908: There are 267 students in faculty and each of them spends approximately 5.625 hours daily in the reading room. Reading room works 8 hours a day. How many seats must be in the reading room that the probability of finding a free seat is not less than 0.78? Use the De Moivre - Laplace theorem
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Determine the total student-hours in the reading room:**
* Total student-hours per day = Number of students * Hours per student per day
* Total student-hours per day = 267 students * 5.625 hours/student = 1503.75 student-hours
**2. Calculate the average student-hours per hour of reading room operation:**
* Average student-hours per hour = Total student-hours per day / Reading room hours per day
* Average student-hours per hour = 1503.75 student-hours / 8 hours = 187.96875 student-hours/hour
**3. Determine the minimum number of seats required:**
* We need to find the minimum number of seats (S) such that the probability of finding a free seat is at least 0.78.
* This means the probability of all seats being occupied is at most 1 - 0.78 = 0.22.
* **Using the De Moivre-Laplace Theorem:**
* The De Moivre-Laplace theorem states that for a large number of independent trials (in this case, the number of students using the reading room at any given hour), the binomial distribution can be approximated by a normal distribution.
* We can model the number of students using the reading room at any given hour as a random variable with a binomial distribution.
* Let:
* X = the number of students using the reading room at a given hour
* n = the number of students (267)
* p = the probability that a student is using the reading room at any given hour (p = 5.625 hours/student / 8 hours/day = 0.703125)
* μ = mean of X = n * p = 267 * 0.703125 = 187.96875
* σ = standard deviation of X = √(n * p * (1 - p)) = √(267 * 0.703125 * 0.296875) ≈ 7.44
* We want to find the minimum number of seats (S) such that:
* P(X > S) ≤ 0.22
* **Standardize:**
* z = (S - μ) / σ
* Find the z-score corresponding to a cumulative probability of 0.78 (since we want P(X > S) ≤ 0.22) in a standard normal distribution table. This z-score is approximately 0.77.
* **Solve for S:**
* 0.77 = (S - 187.96875) / 7.44
* S = 0.77 * 7.44 + 187.96875
* S ≈ 193.6
* **Therefore, the minimum number of seats required in the reading room is approximately 194 to ensure that the probability of finding a free seat is not less than 0.78.**
**Note:** This calculation uses the De Moivre-Laplace theorem as an approximation. For a more precise calculation, you could use the binomial distribution directly. However, with a large number of students (n = 267), the normal approximation should provide a reasonably accurate result.
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