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

Births In a recent year, NYU-Langone Medical Center had 4221 births. Find the mean number of births per day, then use that result to find the probability that in a day, there are 15 births. Does it appear likely that on any given day, there will be exactly 15 births?

The total number of births in a year is equal to 4221.

The total number of births in the year is given to be equal to 4221.

The total number of days in the year is equal to 365.

The mean number of births per day is equal to:

\(\begin{aligned}{c}\mu = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{births}}\;{\rm{in}}\;{\rm{the}}\;{\rm{year}}}}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}\;{\rm{in}}\;{\rm{a}}\;{\rm{year}}}}\\ = \frac{{4221}}{{365}}\\ = 11.56\\ \approx 11.6\end{aligned}\)

The mean number of births per day is equal to 11.6.

Let X be the number of births per day.

Here, X follows a Poisson distribution with mean equal to\({\kern 1pt} \mu = 11.6\).

The probability of 15 births in a day is computed below:

\[\begin{aligned}{c}P\left( x \right) = \frac{{{\mu ^x}{e^{ - \mu }}}}{{x!}}\\P\left( {15} \right) = \frac{{{{\left( {11.6} \right)}^{15}}{{\left( {2.71828} \right)}^{ - 11.6}}}}{{15!}}\\ = 0.0649\end{aligned}\]

Therefore, the probability of 15 births in a day is equal to 0.0649.

Since the probability value is small, it can be said that it is unlikely to have exactly 15 births in a day.