The Birthday Paradox: A Mind-Blowing Probability Trick

The Birthday Paradox is one of those crazy mathematical facts that completely mess with your intuition. It says that in a group of just 23 people, there’s a more than 50% chance that at least two of them share the same birthday. Sounds wild, right? With 365 days in a year, you’d think the chances would be way lower. But let’s break it down and see why this mind-bending probability actually makes sense!

Why Does This Happen?

Our brains naturally think about just one specific person matching someone else’s birthday, which does have a low probability of 1/365. But the trick here is that we’re checking any two people in the group, and that changes everything.

• With 23 people, there are 253 different pairs (yep, the math checks out: 23C2 = 253).

• Each pair has a 1/365 chance of having the same birthday.

• Since there are so many pairs, the odds of at least one match skyrocket!

Breaking It Down Step by Step

Instead of calculating the probability that at least two people share a birthday, let’s flip it and calculate the probability that no one does, then subtract from 1.

1. The first person can have any birthday: 365/365 = 1.

2. The second person must have a different birthday: 364/365.

3. The third person must also have a different birthday: 363/365.

4. Keep going like this for 23 people:

5. When you calculate this, the probability of no match is ~49.27%, meaning the probability of at least one match is:

So, in a room of just 23 people, it’s more likely than not that two will share a birthday! 🎉

How Fast Does This Probability Climb?

As the number of people increases, the probability of a shared birthday jumps up dramatically:

• 23 people → 50.7% chance

• 30 people → 70% chance

• 50 people → 97% chance

• 70 people → 99.9% chance

• 100 people → ~99.99997% chance

• 367 people → 100% chance (Guaranteed match, thanks to the Pigeonhole Principle!)

Why Does 100% Probability Happen at 367 People?

This is where the Pigeonhole Principle comes into play. Since there are only 366 possible birthdays (including leap years), if you have 367 people in a room, someone HAS to share a birthday. No way around it! 🔥

Real-World Uses of This Paradox

This isn’t just a fun fact—it has real applications too!

1. Cybersecurity: Hackers use a concept called birthday attacks to crack hash functions in encryption.

2. Fun Classroom Experiments: Teachers often try this with students, and it’s always surprising!

3. Sports Teams & Events: Large teams or office groups almost always have birthday matches.

Final Thoughts

The Birthday Paradox is one of the coolest ways to see how probability can totally mess with your intuition. While it feels like a birthday match should be rare, math says otherwise. In a group of just 23 people, the odds are already over 50%, and by 50 people, it’s almost guaranteed! 🤯

Next time you’re in a room with a decent-sized crowd, try it out—chances are, someone shares a birthday! 🎂

Names and inventor

Who discovered atom?

John Dalton

Who discovered electron?

J. J. Thomson

Who discovered nucleus of atom?

Ernest Rutherford

Who discovered proton?
Ernest Rutherford

Who discovered neutron?
James Chadwick

Who discovered X-Rays?
Wilhelm Conrad Roentgen

Who discovered Radium?
Pierre Curie and Marie Curie

Who discovered Polonium?
Marie Curie and Pierre

Who discovered Penicillin?
Alexander Fleming






Inventions

Who invented telescope?
Hans Lippershey

Who invented telephone?
Alexander Graham Bell

Who invented thermometer?
Galileo Galilei

Who invented nuclear Reactor?
Enrico Fermi

Who invented injection?
Francis Rynd

Who invented Bulb?
Thomas Edison


Please comment if you want me to add anything to the list and if any mistake has occurred, correct me.

Here is the list of TOP TEN SCIENTISTS
Also, read the blog  Discovery of PI and area of the circle.
Thank you.

Why Nitrogen is the most abundant gas on earth?

The composition of the earth atmosphere is:

Nitrogen: 78%.

Oxygen: 21%.

Other gases: 1%.

Reasons for the abundance of nitrogen on earth's crust are:

Nitrogen is very less reactive with elements present on earth's crust. It does not react to make composites and hence present in the molecular form.

Nitrogen is volatile in nature. So it occurs in the gaseous state.

Now, what is a volatile element?

These elements have a low boiling point so they present in gaseous state at normal temperature and pressure.

It is heavier than other gases. Gases like lithium, helium and hydrogen get escaped from earth to space because of low weight. But, nitrogen has higher molecular weight, thus it does not get escaped. 

Top Ten Scientists of all time

10. Louis Pasteur

27 Dec 1822-28 Sep 1895

Key work: -discoveries of the principles of vaccination, microbial fermentation and pasteurization.

He is remembered for his remarkable breakthroughs in the causes and prevention of diseases, and his discoveries have saved many lives ever since He reduced mortality from puerperal fever, and created the first vaccines for rabies and anthrax. His medical discoveries provided direct support for the germ theory of diseases.

9. James Clerk Maxwell

13 June 1831-5 Nove 1879

-Electromagnetism

-Maxwell’s equation

In the millennium poll – a survey of the 100 most prominent physicists – Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein. His Discoveries helped usher in the era of modern physics, laying the foundation for such fields as special relativity and quantum mechanics.

8. Charles Darwin

12 Feb 1809-19 April 1882

-Wrote the book Origin of Species
-Theory of natural selection
Charles Darwin was an English naturalist, biologist and geologist who contributed to the science of evolution. In his five- year voyage(1831-1836) on HMS Beagle, he observed the data, concluded and wrote the book.

7. Nikola Tesla

10 July 1856-7 Jan1943

-best known for his contributions to the design of the modern Alternating Current

-Tesla coil

6. Galileo Galilei

15 Feb,1564-8 Jan 1642

-He invented thermometer

-His 400 years old telescope is still used by scientists

Galileo is considered as the father of modern science. He is known for his work as astronomer, engineer, mathematician, physicist and philosopher. He studied speed, velocity, freely falling body law of inertia.
Telescope was actually discovered by a Dutch eyeglass maker Hans Lippershey in 1608.  In the next year, Galileo improved his design.

5. Archimedes

287 BC-212 BC

-Archimedes Principle

-Calculated the value of Pi for the first time
Learn how did he calculate the value of Pi

Archimedes is one of the greatest mathematicians of the all time. He invented Archmedes' screw, which is a perfect example of engineering and still used in several places. Galileo Galilei of sixteen century called him superhuman. 

4. Marie Curie

7 Nov,1867-4 July 1934

Key Work:- Invented radium and Polonium
-Discovery of Radioactivity with Henri Becquerel

She won Nobel Prize in Physics in1903 and in Chemistry in 1911. She is the only person who won this prize in two different fields of science. She was also the first women to win this prize. She lost her life because of an explosion of radiations during an experiment. Her hard work and sacrifices for science and humanity are all unforgettable.

3. Michael Faraday

17 Sep 1791-25 Aug 1867

-Albert Einstein kept a picture of Faraday on his study wall, alongside pictures of Isaac Newton and James Clerk Maxwell

-Known for Faraday's law of induction

Electrical Engineering is incomplete without this name.
He established the basic concepts of electromagnetic field and studied the behavior of current.

2. Isaac Newton

4 Jan 1643-31 March 1727

-Gravitational force

-Invented Calculus which is a powerful tool in mathematics

His discovery of Laws of gravitational force is one of the most significant research. It revealed lots of secrets of the universe. It revealed why planets and stars are moving. Why they follow Kepler's Laws. Why things fall on earth. Why the moon does not fall on earth. Newton's contribution to physics and mathematics is matchless.

1. Albert Einstein

14 March 1879-18 April 1955

-Theory of relativity

-Derived the most famous equation E=mc2
-Nobel prize in physics in1921 for discovering the law of photoelectric effect

Albert Einstein is considered as the father of modern physics. he gave the revolutionary contribution to physics. Black Hole, Time Travel, Time Dilation, Relativity were unknown terms before him.

Also Read: How the value of Pi was calculated?

Source: Wikipedia
Youtube Video> Top ten scientists
Note: This information is based on basis of Wikipedia and my own knowledge.
(If you do not satisfied with this list then comment what change should I do)

Pi and Circle

The circle is one of the most important fundamental shapes in geometry. It is being discussed since ancient time. Let we discuss how its area was derived and value of Pi.

Value of Pi was firstly derived by Archimedes 2300 year ago. That time no mathematical tools were available for him. There was no way to write the decimal value. No way to write the rational number and even zero was not found. Let discuss how did he find the value of PI.

He drew two circles of the same size and he inscribed a square inside the first circle. He drew a large square outside the other circle. 

Consider radius of circles is r. 

From the picture, it is clear that the area of the first square is lesser than the area of the circle.

But, area of the second square is greater than that of the circle.

Because both circles have the same radius, therefore, area of both the circles is equal.

From the above diagram, we can conclude that the area of the circle is greater than the area of the first square and lesser than that of the second square. So the value of the amount of area of this circle lies between the areas of these two squares.

Somehow we need to find the areas of both the square by using simple trigonometry.

Area of  first circle=2r²

Area  of second  circle=4r²

Now he did same with pentagon and more polygon.

Area of smaller pentagon =2.37r²

Area of larger pentagon    =  3.6r²

So as we are increasing the sides of the polygon the shape of the polygon is approaching to circle and empty void between circle and polygon is decreasing.

Archimedes did this for polygon having 96 sides.

A polygon with 96 sides looks like a circle and void between circle and polygon is negligible.

  To find the area of square this formula is used

For 96 sides-

Area of larger square=3.143r²

Area of smaller square=3.139r²

Average of both values is considered as the area of the circle which is 3.141r²

This value (3.14) is named as Pi (π)

Some facts about PI:

-14th of March is celebrated as Pi day because 3=3rd month of the yea 14=14th day of March

-Albert Einstein has birthday on 14th March and Stephen Hawking died on 14th March 2018

-Leonardo Euler adopted the symbol for Pi in 1937th