Question

In a recent year, a hospital had 4415 births. Find the mean number of births per day, then use that result and the Poisson distribution 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?

Answer 1

To find the mean number of births per day, divide the total number of births by the number of days in a year. Use the Poisson distribution formula to find the probability of a specific number of births in a day. The probability of exactly 15 births is low.

Step-by-step explanation:

To find the mean number of births per day, we divide the total number of births in a year (4415) by the number of days in a year. Assuming there are 365 days in a year, the mean number of births per day is 4415/365 ≈ 12.10.

To find the probability that in a day there are 15 births, we can use the Poisson distribution. The formula for the Poisson distribution is P(x; λ) = (e^(-λ) * λ^x) / x!, where x is the number of events and λ is the average number of events. In this case, x = 15 and λ = 12.10. We can plug these values into the formula to find the probability.

P(15; 12.10) = (e^(-12.10) * 12.10^15) / 15!

The probability is approximately 0.084. The probability is quite small, so it does not appear likely that on any given day there will be exactly 15 births.