In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

b. Find the probability that in a single day, there are 16 births.

Verified step by step guidance

Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events (e.g., births) occurring in a fixed interval of time or space, given a known average rate of occurrence (λ). The probability mass function is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).

Step 2: Calculate the average number of births per day (λ). Since there are 5942 births in 365 days, divide the total number of births by the number of days to find the daily average: λ = 5942 / 365.

Step 3: Identify the value of k. In this problem, k represents the number of births in a single day, which is given as 16.

Step 4: Substitute the values of λ and k into the Poisson probability formula. Use the formula P(X = k) = (λ^k * e^(-λ)) / k!. Replace λ with the calculated daily average and k with 16.

Step 5: Simplify the expression to compute the probability. First, calculate λ^k, e^(-λ), and k! (16 factorial). Then, divide the product of λ^k and e^(-λ) by k! to find the probability. This will give you the final result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events, such as the number of births in a day.

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For the given problem, λ would be calculated by dividing the total number of births (5942) by the number of days (365), which gives the expected number of births per day.

Calculating Probability with Poisson

To find the probability of observing exactly k events (in this case, 16 births) in a Poisson distribution, the formula P(X = k) = (e^(-λ) * λ^k) / k! is used, where e is the base of the natural logarithm, λ is the mean number of events, and k! is the factorial of k. This formula allows us to compute the likelihood of a specific number of occurrences given the average rate.

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