A specially equipped trauma emergency room at a hospital has been in operation for 40 weeks and has been used a total of 240 times. Assuming the weekly pattern of demand for this facility is Poission, compute the following:
1) The probability that the room is not used in a given week,
2) The probability that the room is used seven or more times in a week, and
3) The mean demand for a two week period.
Poisson distribution with a mean of m is this: P(x) = e^(-m) m^x / x!
For #1: 240/40 = 6 (average per week)
P(0) = e^(-6) 6^0 / 0! = 0.0025 (rounded value)
For #2: P(>=7) = 1 - P(<7) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6)]
You might be able to use a table instead of computing these by hand.
For #3: The mean demand for a two week period = 12.
To calculate the probabilities and mean demand using the given information, we can apply the Poisson distribution formula.
1) The probability that the room is not used in a given week:
We are given that the average usage per week is 6 (240 times in 40 weeks). Using the Poisson distribution formula, we can calculate P(0), which represents the probability that the room is not used in a week.
P(0) = e^(-6) * 6^0 / 0! = e^(-6) * 1 / 1 = e^(-6) ≈ 0.0025 (rounded to four decimal places)
2) The probability that the room is used seven or more times in a week:
To calculate this probability, we need to sum the probabilities of the room being used 7, 8, 9, and so on, up to infinity. However, this can be quite time-consuming to compute by hand. Instead, we can use the complement rule to calculate the probability of the room being used seven or more times.
P(≥7) = 1 - P(<7)
Here, P(<7) represents the sum of probabilities for the room being used 0, 1, 2, 3, 4, 5, and 6 times in a week. We can subtract this sum from 1 to get P(≥7).
3) The mean demand for a two-week period:
The average usage per week is 6. Therefore, the average usage for a two-week period would be twice that value, i.e., 2 * 6 = 12.
By applying the formulas and calculations described above, we can find the desired probabilities and mean demand for the given scenario.