3. At Kabwe General Hospital, an average of 6 accident victims arrive per night in the intensive care unit. However, arrangements have been made such that the intensive care unit is able to handle 8 accident victims per night. If this number is exceeded the patients will not get optimal treatment as the unit’s capacity will have been exceeded. For the purpose of this exercise, let’s assume that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed with λ = 6. a) Is the assumption that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed a valid one? Give augments in favour and against this assumption. b) Given a night, what is the probability that the unit’s capacity will be exceeded? c) Given 10 nights, what is the probability that the capacity of the unit will be exceeded? d) What is the probability of at most 8 accident victims arriving at the unit in a given night?
Question
- At Kabwe General Hospital, an average of 6 accident victims arrive per night in the intensive care unit. However, arrangements have been made such that the intensive care unit is able to handle 8 accident victims per night. If this number is exceeded the patients will not get optimal treatment as the unit’s capacity will have been exceeded. For the purpose of this exercise, let’s assume that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed with λ = 6. a) Is the assumption that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed a valid one? Give augments in favour and against this assumption. b) Given a night, what is the probability that the unit’s capacity will be exceeded? c) Given 10 nights, what is the probability that the capacity of the unit will be exceeded? d) What is the probability of at most 8 accident victims arriving at the unit in a given night?
Solution
a) The assumption that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed can be considered valid if the following conditions are met:
- The number of accident victims arriving at the hospital is independent of the time since the last arrival.
- The probability of an accident victim arriving is the same for any two equal periods of time.
Arguments in favor of this assumption could be that accidents are random events and can occur independently of each other. Arguments against this assumption could be that there may be factors such as weather conditions, day of the week, or time of the day that could affect the rate of accidents and hence the arrivals at the hospital.
b) The probability that the unit’s capacity will be exceeded can be calculated by finding the probability of more than 8 accident victims arriving at the unit in a given night. This can be calculated using the Poisson distribution formula:
P(X > 8) = 1 - P(X ≤ 8)
c) Given 10 nights, the probability that the capacity of the unit will be exceeded at least once can be calculated by finding the probability of more than 8 accident victims arriving at the unit in a given night and raising it to the power of 10.
P(X > 8 in 10 nights) = [1 - P(X ≤ 8)]^10
d) The probability of at most 8 accident victims arriving at the unit in a given night can be calculated using the Poisson distribution formula:
P(X ≤ 8) = Σ [e^(-λ) * (λ^x) / x!] for x = 0 to 8
where λ = 6 (average number of accident victims per night), e is the base of the natural logarithm (approximately equal to 2.71828), and x! is the factorial of x.
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