.Arrivals to a certain hotel follow a Poisson process. The expected number of arrivals each week is 4. What is the probability that there are exactly 3 arrivals over the course of one week?
To solve this problem, we can use the Poisson probability formula. The Poisson process is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space.
The formula for the Poisson probability is given by:
P(x;λ) = (e^(-λ) * λ^x) / x!
Where:
P(x;λ) is the probability of x events occurring in a given time interval,
e is Euler's number, approximately equal to 2.71828,
λ is the average rate of events occurring in the time interval, and
x is the desired number of events.
In this case, the average rate of arrivals each week is 4, so λ = 4. We want to find the probability of exactly 3 arrivals, so x = 3. Plugging these values into the formula:
P(3;4) = (e^(-4) * 4^3) / 3!
Now, let's calculate this probability:
P(3;4) = (e^(-4) * 64) / 6
Calculating e^(-4) ≈ 0.01832:
P(3;4) ≈ (0.01832 * 64) / 6
P(3;4) ≈ 1.852 / 6
P(3;4) ≈ 0.3086
Therefore, the probability that there will be exactly 3 arrivals at the hotel in one week is approximately 0.3086.