Question

A small college has 1460 students. What is the approximate probability that more than six students were born on Christmas day? Assume that birthrates are constant throughout the year and that each year has 365 days.

Answer 1

The approximate probability that more than six students were born on Christmas day is P=0.105.

Explanation:

This can be modeled as a binomial variable, with n=1460 and p=1/365.

The sample size n is the total amount of students and the probability of success p is the probability of each individual of being born on Christmas day.

As the sample size is too large to compute it as a binomial random variable, we approximate it to the normal distribution with the following parameters:

`\mu=n\cdot p=1460\cdot (1/365)=4\\\\\sigma=√(n\cdot p(1-p))=√(1460\cdot(1/365)\cdot(364/365))=√(3.989)=1.997`

We want to calculate the probability that more than 6 students were born on Christmas day. Ww apply the continuity factor and we write the probability as:

`P(X>6.5)`

We calculate the z-score for X=6.5 and then calculate the probability:

`z=(X-\mu)/(\sigma)=(6.5-4)/(1.997)=(2.5)/(1.997)=1.252\\\\\\P(X>6.5)=P(z>1.252)=0.105`