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The Birthday Paradox

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by

Isra Ahmed

on 3 January 2014

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Transcript of The Birthday Paradox

The Birthday Paradox
What is a paradox?

A paradox is a situation which, because of its existence, indicates that it shouldn't exist. It is a statement that appears to be self-contradictory or silly but may include an unexpressed truth.
An Example: The Chicken or the Egg?!?!
A really famous paradox is, "Which came first? The chicken or the egg?"

Why is it a paradox?
You need a chicken to lay an egg but you need an egg in order for a chicken to hatch. So, where did the first chicken come from to lay the first egg or where did the first egg come from to hatch the first chicken?
You can't have the first chicken without the egg and you can't have the first egg without the chicken. According to this paradox, there shouldn't be any chickens in the world today.
There are 50 people in a room, I bet that at least two will have the same birth day and birth month.
Would you take the bet?
The idea that if there are 23 people in a room, there is a 50-50 chance that two of them will have the same birthday.

The Birthday Paradox
Most people find this surprising. But even more surprising is the fact that with 50 people, the probability is 97% and with 100 people, the chances are better than 3,000,000 to one that at least two people will have the same birthday.
How is it a paradox?
I have no idea. Google failed me. :(
it could be because the chances of two people having the same birthday shouldn't be so high when the people are so low. For example, when there are 23 people, there is a 50 % chance, yet there are 365 days in the year.
How Does it Work?
Step 1 : Stop being Selfish!
When looking at this problem, a person will automatically assume that at least one person in the room has to have the same birthday as them. But you aren't the only one in the room! there are other people and it could be any of them who have the same birthday as anyone else from the room.
Assuming for a moment that birthdays are evenly distributed throughout the year, if you're sitting in a room with fifty people in it, what are the chances that two of those people have the same birthday? A reasonably, intelligent person might point out that the odds don't reach 100% until there are 366 people in the room (the number of days in a year + 1)... and fifty is about 13.7% of 366... so such a person might conclude that the odds of two people in fifty sharing a birthday are about 14%. However...
In a group of n people, what is the chance of at least 2 of them having the same birthday? and by birthday, it means the same month and day.
The Question:
Let's Make it Easier
In a group of n people, what is the chance of them all having different birthdays?
New Question:
Does it still help us solve the original question?
In this problem, either everyone has a different birthday, or there is at least one match
Chance(Different) + chance(Match) = 1
We can rearrange the formula so that:
Chance(Match) = 1 - Chance(Different)
Let's Start With a Simple Problem
Suppose there are 5 people.
We want to find all the possible ways the people can have a different birthday (not including February 29)
the first person can have a birthday on any of the 365 days
the second person can have a birthday on any of the other 364 days
the third person can have a birthday on any of the other 363 days
the forth person can have a birthday on any of the other 362 days
the fifth person can have a birthday on any of the other 361 days

The problem becomes:
Possible ways 5 people can have DIFFERENT birthdays
= 365 x 364 x 363 x 362 x 361
Now, lets think of how many ways they CAN have birthdays
-they can each have one on any day
Possible ways 5 people CAN have birthdays
= 365 x 365 x 365 x 365 x 365
Total ways 5 people can have DIFFERENT birthdays
Total ways 5 people CAN have birthdays
=
Chance(Different)
Chance(Different)
365 x 364 x 363 x 362 x 361
365 x 365 x 365 x 365 x 365
=
=
0.97
Chance(Match) = 1 - Chance(Different)
Chance(Match) = 1 - 0.97
= 0.03
therefore, in a group of 5 people, there is a 3 % chance of two people having the same birthday
So, what is the chance of two people having the same birthday in a room of 50 people?
Recall, the formula was:
Chance(Match) = 1 - Chance(Different)
and
Chance(Different)
=
Total ways 5 people can have DIFFERENT birthdays
Total ways 5 people CAN have birthdays
Chance (Match)
=
365 x 364 x 363 x ... (365 - n + 1)
365^^n
Chance (Match)
=
365 x 364 x 363 x ... (365 - 50 + 1)
365^^50
1 -
1 -
If you do all the calculations
Chance (Match) =
1 - 0.03
= 0.97 or 97%
In the beginning of the presentation, I said ...
If you said yes, then
This is the Birthday Paradox.
So, What is the chance of two people having the same birthday in our classroom?
We have I think, 26 people in our class
chance(Match) =
365 x 364 x ...(365 - 26 + 1)
365^^26
Remember, sometimes, a calculator is your best friend!
Chance (Match) =
1 - 0.41
= 0.59
In our class, there is about 60% chance that at least 2 of us in the whole room have their birthday on the same day.
Full transcript