The Birthday Paradox

The birthday paradox states that there is 50% of chance that the two people will share the same birthday when 23 people are randomly picked and placed in some room. If there is a group of 30 people then there is a chance of 70%. Not only this, you only need only 70 people to get a chance of 99.9% that atleast two of them will have the same birthday and on the other hand there is never absoulte guarantee of how many people you may fit in a room, even the entire earth's population, take everything insects, plants, animals there is always a chance that no two individuals in that room will have the same birthday as yours. Today we are going to prove the above statements using one of the interesting and my favourite branch of mathematics "PROBABILITY".

So lets analyse the paradox case by case :

Case 1: (50% chance)

Consider an empty room and a random guy came in, what is the probability that two guys share the same birthday, well in this case 0 because there is no one in the room or we can say he will have 365 possible birthday hence his chances of having a unique birthday are absolute P(1) = 365/365, when the second guy came, now we will see the probability of having a different birthday, so for the second guy to have the different birthday than the first one he had 364 choices because one had already been taken by the first one, I will denote the probabilites with P(x) where x is the number of people already in the room and P(x) is the probability for the

So we can say that P(2) = 364/365

For the third guy, we are going to have only 363 possible birthdays which are different from the previous two, so we can say that P(23) = 343/365.

and we can continue the process and in general we can see P(x) = (365-x+1)/365

P(4) = 362/365,

P(30) = 343/365.

Now we times the above probabilities to get the chances of overall happening of the event in a room full of 23 people, it can be represented as :

Now we subtract the probability of unique birthday for everyone from 1 in order to get the probability of having the same birthday for atleast two persons in the room

So we have approximately (1-0.491)*100 = 50.9% hence there is 50% chance for in a room of randomly selected 23 individuals.

This means that if we had 100 rooms fill of 23 random people then out of them there will be 50 rooms which will have atleast two individuals with same birthday and other 50 will not.

There is one more complicated way to do it, let's take 23 people in a room, now the probability of having same birthday for atleast 2 people will be equal to probability of having same birthday for exactly 2 people + probability of having same birthday for exactly 3 people + probability of having same birthday for exactly 3 people +.....+ probability of having same birthday for exactly 23 people, lets denote them as P(i), i being the number of people having the same birthday.

I will explain the above expressions with one examples and then we can observe the pattern and generalize it.

Suppose there are 23 people in the room and we need to calculate the chances of same birthday for exactly two people. First there will be 23C2 ways to choose 2 people out of 23 and we need to work out for probability for each combination and sum them but since they will be equal we just say 23C2 times the probability for any 2 combination, now what should be the probability for a particular combination, as there are 365 days in a year (Consider the year is not leap), we can say there are 1/365 chances for a person's birthday to fall on 1st Jan and similarly for the second one, but simultaneously their chances reduces to 1/365 * 1/365 and now we can say the same for each day in a year so the chances increases to 365*(1/365)*(1/365).

Hence we will get 23C2 times 1/365 as final result.

Now we can see the pattern and generalize the same, we will get the above formula.

So that was another complicated way to work out the same, in this case we need not subtract the probability from 1 as it is already the probability of atleast two individuals having the same birthday.

Case 2: (99.9% chance)

Repeat the same as done in case one, the value of i becomes 70, hence our formula will look like this :

Now subtract the above result from 1 we will get 0.99, hence the chances for atleast 2 people having the same birthday are 99.9% when the total person in the room are 70.

Case 3: (50% chances for matching birthday with yours)

Now let's change the problem a little bit. Now we will concentrate on your birthday, let us suppose you are inside a room full of 23 people including yourself, now how many people are required that there will be 50% chance of someone's birthday being matched to yours.

Now, the calculation for this is not as simple as previous cases, it follows as shown below :

See the difference, earlier we need only 23 people and now we need 254 to get 50% lucky for the same birthday, this difference arises because earlier we were looking for all the possible combinations of people but now it is only us and we have a fixed birthday, hence the chances get lowered.

Case 4: (99.9% chances for matching birthday with yours)

We can repeat the same procedure as for the previous case only the value 50% replaced by 99.9% here is the calculation :

So we need atleast 2519 people in order to get a 99.9% chance of getting atleast one birthday twin, but what about absolute guaranteed, will it be possible to get a 100% assurance that you will get a birthday twin, lets find out.

Case 5: (100% chances for matching birthday with yours)

Do the same steps again and lets see what happens

We will encounter log(0) and we know that log(0) is not defined, hence we can say that the case of absolute probability is not possible here.

But what is happening, Why is that log(0) appearing?

To answer is very simple, no matter how many people you gather, the whole state, whole country or the whole earth's population there will always a chance that your birthday will never match with anyone else.

Its even possible though highly unlikely that all the person have a same birthday which is different from you or may be same as you who knows that's why we call it "PROBABILITY".

Let's make some mathematical observations of the above expression:

We can see that below x = 1, the logarithmic value increases(towards -ve) very fast, but it meets the y-axis at infinity or we can say never that expression which contains log(t) where t->0, will not be defined or we can say we need infinite amount of people to get absolute guarantee for your birthday twin.

So, is all that stuff a paradox, the answer is no, its just mathematics!

Thanks for reading folks, hope you enjoyed the article, see you again!