To solve this problem, we can use the Poisson distribution. The Poisson distribution is commonly used when we want to calculate the probability of a certain number of events occurring within a specific time period, given the average rate of occurrence.
The Poisson distribution is defined as:
P(X = k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!}
where:
- P(X = k) is the probability of k events occurring
- \lambda is the average rate of events occurring in the given time period
- e is the base of the natural logarithm
- k! is the factorial of k, which represents the number of ways to arrange k events
In this problem, the average rate of babies born per hour is 6 ( \lambda = 6 ) and we want to find the probability of three babies being born in a particular 1-hour period ( k = 3 ).
Using the Poisson distribution formula, we can calculate the probability:
P(X = 3) = \frac{6^3 \cdot e^{-6}}{3!}
Simplifying further:
P(X = 3) = \frac{216 \cdot e^{-6}}{6}
Calculating the value of e^{-6} :
e^{-6} \approx 0.00248
Substituting this value into the probability formula:
P(X = 3) = \frac{216 \cdot 0.00248}{6}
Simplifying further:
P(X = 3) \approx 0.089
Therefore, the probability that three babies are born during a particular 1-hour period is approximately 0.089.
Answer: \approx 0.089