Question 1191373: Find each Poisson probability, using a mean arrival rate of 10 arrivals per hour.
(c) Fewer than five arrivals. (Round your answer to 4 decimal places.)
Poisson probability:
(d) At least 11 arrivals. (Round your answer to 4 decimal places.)
Poisson probability:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the Poisson probabilities:
**Understanding the Poisson Distribution**
The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event. The formula is:
P(x) = (e^(-λ) * λ^x) / x!
Where:
* P(x) is the probability of x events occurring
* e is the base of the natural logarithm (approximately 2.71828)
* λ is the average number of events (arrivals) per interval (10 in this case)
* x is the number of events we're interested in
* x! is the factorial of x
**c) Fewer than five arrivals (x < 5):**
We need to calculate P(0) + P(1) + P(2) + P(3) + P(4).
* P(0) = (e^-10 * 10^0) / 0! ≈ 0.000045
* P(1) = (e^-10 * 10^1) / 1! ≈ 0.000454
* P(2) = (e^-10 * 10^2) / 2! ≈ 0.002270
* P(3) = (e^-10 * 10^3) / 3! ≈ 0.007567
* P(4) = (e^-10 * 10^4) / 4! ≈ 0.018918
P(x < 5) = P(0) + P(1) + P(2) + P(3) + P(4) ≈ 0.000045 + 0.000454 + 0.002270 + 0.007567 + 0.018918 ≈ 0.029254
Rounded to 4 decimal places, the probability is 0.0293.
**d) At least 11 arrivals (x ≥ 11):**
It's easier to calculate the complement and subtract from 1. That is:
P(x ≥ 11) = 1 - P(x < 11) = 1 - [P(0) + P(1) + ... + P(10)]
Calculating each individual probability from P(0) to P(10) and summing them can be tedious. It's best to use a calculator or statistical software that has built-in Poisson cumulative distribution functions (CDF).
Using a calculator or software, you'll find that P(x < 11) ≈ 0.5830.
Therefore, P(x ≥ 11) = 1 - 0.5830 = 0.4170
Rounded to 4 decimal places, the probability is 0.4170.
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