The Birthday "Paradox" Problem 🍰🎉

The concept of probability may seem like a very straightforward and obvious if it is seen from the surface level. However, if you go to the depth of the concept, you can find a little level of difficulty due to several mathematical operations. Nevertheless, if you succeed in grasping the basic concept of probability, it is no longer a tedious task. Once you understand it, it may seem tricky and challenging but quite interesting to you. You might have solved problems generally relating to rolling a die, deck of cards, colored balls, tossing a coin etc. Here, we are going to explain something different yet interesting.

            

In this blog, we will discuss a “paradox” of birthday problem probability which will precisely make you familiar about the compounding power of paradox which is strange, counter-intuitive but factually true! Generally, this problem is viewed as counter-intuitive because it goes against our intuition as our brains are bad at figuring out the power of chance. Our brains have trouble estimating the things that grow exponentially because we are not accustomed to such calculations in our daily life. We generally expect probabilities to be linear as we consider the conditions only through which we are familiar. We don’t think of other possibilities. This limits our thinking and we find exponential probabilities to be counter-intuitive.

            

Many of you wait the whole year for your birth date, right? But if you come to know that your colleagues share the same birthday as you, you become even more excited to bloom your birthday candles and celebrate the day. So, let’s begin!

The birthday problem (also called the birthday paradox) deals with probability that in a set of $$n randomly selected people, at least two people share the same birthday. It states 3 points-

·       a)The birthday problem states that the probability of at least 2 people sharing the same birthday is 50% when total number of people are 23.

·        b)Probability of at least 2 people sharing the same birthday is 99% if total number of people are 70.

·        c)Probability of at least 3 people sharing the same birthday is 100% when n >365.

 We will elaborate on first point by which you can easily by yourself understand the rest two points. The following explanation will explain you about correct way of thinking about the problem and the reasoning behind it.

 Standard assumptions

To make things simpler we make some standard assumptions –

There is no leap year.

There is no twin born.

Every event is equally likely .i.e.  Seasonal, or weekday variations are disregarded.

At first glance, many people guess 183 for not so obvious reason that it is half of the number of days in the year, which seems quite intuitive. Unfortunately, such intuition doesn’t take mathematical tool in solving this problem and rather you will in the birthday paradox TRAP.

 

Let's find why ?

With 23 people in a room, the number of pairs in the room can be taken out through combination,

 

 

Where n = 23 and r = 2

We know that, the chance or probability of 2 people having different birthdays in a year of 365 days is

This seems easy, right? This is because when we consider 2 people, we have only 1 pair.

But things get complicated with 253 pairs because of extraneous information. So we make use of exponents to find the probability:

P (at least two people share same birthday) = 1 — 0.4995= 0.50005

                                                         =  50.005% APPROX

The easy intuitive trick that solves the birthday problem!

Instead of using the not “so easy” exponents rule, we can correct but the intuitive way to easily solve the birthday paradox.

P (At least one pair shares birthday) = 1 – P (All birthday are on different days)

So, now we focus our energy to find the answer to this: “What’s the probability of people NOT any birthday in the group?”

Unique birthdays DENOMINATOR

Intuitively we know, the denominator is 365ⁿ (where n = 23). The denominator is unscathed.

Unique birthdays NUMERATOR

This is where the MAGIC BOOM happens!

The first folk in the group has 365 options for a birthday (GREEDY BOAR). This leaves the second folk with only (365-1) option, as we force them to select a different birthday.

In this correct but initiative manner, we can say that the N person has (366— N).

If N is 23, we put in the multiplication signs to get the numerator,   365 x 364 x 363 x 362 x 361 x 360 x 359 x 358 x 357 x 356 x 355 x 354 x 353 x 352 x 351 x 350 x 349 x 348 x 347 x 346 x 345 x 344 x 343 = …

 P(All 23 birthdays unique) = (365 x 364 x 363 x 362 x 361 x 360 x 359 x 358 x 357 x 356 x 355 x 354 x 353 x 352 x 351 x 350 x 349 x 348 x 347 x 346 x 345 x 344 x 343) / (365 ^ 23)

= 0.4995 (rounded off to the capacity of my attention span)

Finally, All’s well that ends well

P (at least two people share same birthday) = 1 — 0.4995= 0.50005

                                                                      = 50.005% APPROX