In a study for a local clinic, you have found that the average number of patients between 9am and 12noon on Friday is five patients. Using a Poisson distribution and the probability more than eight patients will arrive?
To use the Poisson distribution to calculate the probability of more than eight patients arriving, you need to know the lambda value, which represents the average number of events (in this case, patients) occurring in a given time period. In this scenario, the lambda value is given as five patients.
The Poisson probability mass function is given by the formula:
P(X = k) = e^(-λ) * (λ^k) / k!
where P(X = k) is the probability of having exactly k events, λ is the average number of events (five patients in this case), e is Euler's number (approximately 2.71828), and k is the number of events.
To find the probability of more than eight patients arriving, you need to calculate the probabilities of having eight or fewer patients and subtract it from 1 (as the sum of all probabilities must equal 1).
Let's calculate the probability step by step:
1. Probability of having eight or fewer patients:
P(X ≤ 8) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 8)
You can use the formula mentioned earlier to calculate each individual probability and sum them up.
2. Probability of more than eight patients:
P(X > 8) = 1 - P(X ≤ 8)
By following these steps, you can calculate the probability of more than eight patients arriving using the given average number of patients (lambda) and the Poisson distribution.