Answer :
Final answer:
In the Poisson distribution model, given a rate of 2 arrivals every 15 minutes, the probability of 4 arrivals in the next 15 minutes is 0.009.
Explanation:
This question deals with the Poisson distribution, a concept in probability theory that describes the probability of a given number of events occurring in a fixed interval of time or space. In this case, the 'event' is the arrival of patients to the emergency department of a hospital, and the 'interval' is 15 minutes.
First, note that the rate, λ (lambda), is the average number of arrivals per interval, which is 2 in this case. The formula for the Poisson probability is P(X=k) = λ^k * e^(-λ) / k!, where e is the base of natural logarithms and k is the specific number of occurrences you are solving for, which in this case is 4.
Substitute 4 for k and 2 for λ in the formula: P(X=4) = 2^4 * e^(-2) / 4! = 2.4 * 0.135 / 24 = 0.009.
So, the probability that there will be 4 arrivals in the next 15 minutes, if arrivals follow the Poisson model, is c) 0.009. This is a very low probability, indicating that it is highly unlikely for there to be 4 arrivals in the next 15 minutes if the average rate of arrivals is 2 every 15 minutes.
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