Question 1168310: On average there are three babies born a day at the Hospital with hair on their backs. Find the probability that in one day no babies are born with hair on their backs.
Round answer to 4 decimal places.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! This problem can be solved using the Poisson distribution, which 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.
**1. Understand the Poisson Distribution**
* The Poisson distribution formula is: P(X = k) = (e^(-λ) * λ^k) / k!
* P(X = k) is the probability of k events occurring.
* λ (lambda) is the average number of events (in this case, babies born with hair on their backs).
* e is Euler's number (approximately 2.71828).
* k! is the factorial of k.
**2. Identify the Values**
* λ = 3 (average number of babies born with hair on their backs per day)
* k = 0 (we want to find the probability of no babies born with hair on their backs)
**3. Apply the Poisson Formula**
* P(X = 0) = (e^(-3) * 3^0) / 0!
* P(X = 0) = (e^(-3) * 1) / 1
* P(X = 0) = e^(-3)
**4. Calculate the Probability**
* e^(-3) ≈ 0.049787
**5. Round to 4 Decimal Places**
* P(X = 0) ≈ 0.0498
**Therefore, the probability that no babies are born with hair on their backs in one day is approximately 0.0498.**
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