Find the binomial probability for the given problem, and...

Find the binomial probability for the given problem, and then compare the normal and the Poisson approximations.
Out of 1095 people, what is the probability that exactly 2 were born on Jan. $1 ?$ Assume 365 days in a year.

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by its parameters n (number of trials) and p (probability of success in each trial), and its probability mass function calculates the probability of achieving exactly k successes. In this context, it is used to determine the likelihood of exactly a certain number of people having a specific birthday out of a large population.

Normal Approximation

The normal approximation is used to estimate the probabilities of a binomial distribution when the number of trials is large and both np and n(1-p) are sufficiently large. Under these conditions, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p), making the computation of probabilities more manageable. This approach simplifies the calculation when dealing with high sample sizes.

Poisson Approximation

The Poisson approximation is a method used to approximate the binomial distribution when the number of trials is large and the probability of success is very small, such that the average number of successes (np) remains moderate. This approximation uses the Poisson distribution with parameter ? = np, and it is particularly useful in situations involving rare events over many trials, enabling easier probability calculations.

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