# I. the birthday paradox (also known as

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).

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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

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

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.