Math Practice Ia

Math Practice IA

An Investigation of the Application of the Birthday Paradox

 Emma Hope -  DATE \@ "MMMM d, y" April 14, 2015

Introduction

How large must a sample be to make the probability of finding two people with the same birthday at least 50%?

What is the probability that, in a set of randomly chosen people, any given pair will

share the same birthday?

We assume the following:

1. There are 365 days in a year
2. There are no twins in the sample

To solve this problem we must first consider the basic rules of probability: the sum of the probability that an event will happen and the sum of the probability that an event will not happen is 1. Thus, there is a 100% chance that said event may or may not happen. By using this logic, we are able to determine the probability that no two people in a sample will share a birthday and therefore [pic 3]determine the probability that two people will share a birthday.

P(event will happen) + P(event will not happen) = 1

P(two people will share a birthday)) + P(two people will not share a birthday) =1

P(two people will not share a birthday) = 1- P(two people will not share a birthday)

Due to the fact that there are 365 days in a year, Person 1 can have any birthday and Person 2 must have a different birthday. So the probability of this is 364/365. For Person 3 there is a 363/365 chance for them to have a different birthday. To find the probability that Person 2 and Person 3 have different birthdays:

(365/365) x (364/365) x (363/365) ≈ 99.18%

So if we wanted to know the probability of 4 people having different birthdays, we do the same equation.

(365/365) x (364/365) x (363/365) x (362/365) ≈ 99.18%

This formula can be used for up to 364 people. A formula for the probability of n people having different birthdays would be:

((365-1)/365) x ((365-2)/365) x ((365-3)/365) x . . . x ((365-n+1)/365).

To calculate our original question,