I.

Introduction

Have

you ever wondered how many people, if any, in a room shared a birthday? Well,

how many people do you think it would take to find two people who share the same

birthday? In fact, if you were to get the birthdays of 23 random people, there would

be a 50 percent chance that two of them would share the same birthday. Now if

we have a bigger group, such as 75 people, there would be a 99.9% chance that

at least two of them would have matching birthdays. This whole concept is actually

the birthday paradox (also known as the birthday problem).

I

became interested with the birthday paradox one day when my seventh-grade math

teacher had briefly explained it to us and sounded very simple. After the

birthday paradox was explained to us, my teacher wanted to test it out. It

worked out perfectly because we had exactly 23 people in the room minus the teacher.

Once we wrote all of our birthdays on a note card, our teacher collected them

and started to organize them by day and month. Once they were all set out, we

quickly discovered that two students in the class had the same birthday. It all

just seemed to be a coincidence. At the time, I didn’t really understand the

math behind it, although the whole idea of the birthday paradox really

fascinated me. Now that I am older, I am able to take matter into my own hands

and further my knowledge of the mathematical concept behind the birthday paradox.

I will do this by conducting my own experiment and using the birthdays of my

peers.

II.

Birthday

Paradox Background

The

birthday paradox tells us that with a group of 23 people, there is a 50% chance

that two people will share the same birthday. But how could this possibly even

be accurate? There seems to be several explanations why many find this to be a paradox.

So let’s say there’s 23 people in one room. If one person from that group of 23

tries to compare their birthday with someone else, it would then make for 22

comparisons since that leaves only 22 chances for people to share the same

birthday. When

the 23 birthdays of people in the group are compared against each other, there’s

more than the 22 birthday comparisons, meaning the 1st individual

would have 22 comparisons to make, and because the 2nd individual

had already been compared to the 1st, there would be 21 comparisons to

make and so forth. Afterwards, if you add all possible comparisons, which is 22

+ 21 + 20 + … +1, the sum you would end up getting would be 253 comparisons. Therefore,

this shows that in a group of 23 people, there will be 253 possibilities to

having a shared birthday.

III.

Proving

the Birthday Paradox

Experiment

Materials

that I will need to conduct this experiment will be:

·

6 groups of 23 people, 6 groups of 75 people;

588 people total.

·

Notecards

Procedure:

I will prove the birthday

paradox with 12 different groups. Half of them will contain 23 people, while

the other half contain 75 people. It is being split up this way so that we can

see if the birthday paradox is true. With the 6 groups of 23 people, will half

of the groups have at least 2 people who have the same birthday? What about the

other 6 groups that have 75 people? Will they have a 99.9% chance that 2 people

share the same birthday? After, I will write the out every person’s birthday on

a notecard. Once completed, I will shuffle all the cards and start creating the

groups of 23 and 75. After all the cards are distributed in their groups, I

will collect the data of the matching pairs.

When

comparing probabilities with birthdays, it can be easier to look at the

probability that people do not share a birthday. A person’s birthday is one out

of 365 possibilities (excluding February 29 birthdays). The probability that a

person does not have the same birthday as another person is 364 divided by 365

because there are 364 days that are not a person’s birthday. This means that

any two people have a 364/365, or 99.726027 percent, chance of not matching

birthdays.

As

mentioned before, in a group of 23 people, there are 253 comparisons that can

be made. So, we’re not looking at just one comparison, but at 253 comparisons.

Every one of the 253 combinations have the same odds, 99.726027 percent of not

being a match. If enter (364/365)253 into your calculator, it’s shown that there’s

a 49.95% chance that the 253 comparisons don’t have matches. Because of that,

the chances that there are similar birthdays with the 253 possible comparisons

is 1 minus 49.952%, which equals to 50.048%. So, the additional trials that

there are, the closer the actual probability comes to 50 percent.

IV.