Births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability of obseeving  4 births in a given hour at the hospital

To solve this problem, we use the **Poisson distribution**, which is appropriate for modeling the number of times an event occurs within a fixed interval of time or space, provided these events happen with a known constant mean rate and independently of the time since the last event.

Given:
- The average rate of births, \(\lambda\), is 1.8 births per hour.
- We want to find the probability of observing \(k = 4\) births in a given hour.

The probability mass function (PMF) of the Poisson distribution is given by:

\[
P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}
\]

Substituting the given values:

\[
P(X = 4) = \frac{e^{-1.8} \times 1.8^4}{4!}
\]

Calculating step-by-step:

1. **Calculate \(e^{-1.8}\)**:
   \[
   e^{-1.8} \approx 0.1652989
   \]

2. **Calculate \(1.8^4\)**:
   \[
   1.8^4 = 1.8 \times 1.8 \times 1.8 \times 1.8 = 10.4976
   \]

3. **Calculate \(4!\) (factorial of 4)**:
   \[
   4! = 4 \times 3 \times 2 \times 1 = 24
   \]

4. **Substitute all values into the PMF**:
   \[
   P(X = 4) = \frac{0.1652989 \times 10.4976}{24}
   \]

5. **Final calculation**:
   \[
   P(X = 4) = \frac{1.73460657}{24} \approx 0.0722753
   \]

Therefore, the probability of observing 4 births in a given hour at the hospital is approximately **0.0723** or **7.23%**.

If you want more details or have any questions, feel free to ask. Here are some related questions you might be interested in:

1. What is the probability of observing exactly 2 births in a given hour?
2. How do we calculate the probability of observing no births in an hour?
3. What is the probability of observing more than 3 births in a given hour?
4. How does the Poisson distribution change with different average rates (\(\lambda\))?
5. Can we use the Poisson distribution for a scenario with a variable average rate?
6. What is the expected number of births in a 2-hour interval?
7. How does the Poisson distribution differ from the binomial distribution?
8. How would the calculations change if we were interested in births per day instead of per hour?

**Tip**: Understanding the relationship between the average rate of an event and the interval over which it is measured is key when applying the Poisson distribution.

Calculate the probability of observing 4 births in a hospital hour using the Poisson distribution with an average rate of 1.8 births per hour. Step-by-step solution included.

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