How Many Strangers Does It Take in a Room for Two to Share the Same Birthday?

Imagine you’re at a birthday party, the room alive with laughter and chatter as you stand before the cake, ready to cut the first slice. As the knife touches the cake, your mind wanders what are the chances that someone else in this room shares the same birthday as you? With 365 days in a year, it might seem like there’s only a 1 in 365 chance, meaning you’d need 365 people in the room for it to be likely. But what if the reality was much different? What if, based on mathematical principles, a much smaller group of people could make this surprising coincidence happen? The answer might just surprise you. This is known as the Birthday Paradox.

In 1939, Richard von Mises, an Austrian-born mathematician and engineer, introduced the concept of the “Birthday Paradox.” He was widely recognized for his contributions to statistics and probability theory. His career was diverse, he taught aviation, designed airfoils, and even built a plane for the Austrian army during World War I. After fleeing Nazi Germany in 1933, he continued his work at the University of Istanbul and later at Harvard University, where he co-founded Advances in Applied Mechanics, a leading journal in the field.

Von Mises made key contributions to probability theory, including the axiom of convergence, which states that the relative frequency of an event approaches a limit as more observations are made. He also introduced the axiom of randomness, which asserts that this limit remains constant regardless of the chosen sequence. These ideas were published in his book Probability, Statistics and Truth (1928). However, his attempt to define probability objectively faced philosophical criticism, particularly for its positivistic approach.

If the concept of probability and the formulae of the theory of probability are used without a clear understanding of the collectives involved, one may arrive at entirely misleading results. – Richard von Mises

Cracking the Birthday Paradox!

However, Von Mises solved the birthday paradox with a straightforward probabilistic calculation, revealing that only 23 people are required in a room for the probability to surpass 50% that two people will share the same birthday.

Von Mises made these assumptions first

The year has 365 days (excluding leap years).
Each person’s birthday is equally likely to fall on any of these days.
Birthdays are independent of each other.

To determine the probability of at least two people sharing a birthday, he considers the reverse scenario, the probability that all individuals have distinct birthdays. Then subtract this value from 1 to find the probability of at least one shared birthday.

Mathematically, if P(n) is defined as the probability that n people all have unique birthdays, he calculates it as follows:

Once computed P(n), the probability that at least two people share a birthday is:

Here’s a step-by-step guide on how to solve the birthday paradox,

Step 1: The First Individual

The first person enters the room,no constraints exist. Their birthday may fall on any of the 365 days.

Step 2: The Second Individual

A second person arrives. For them not to share a birthday with the first, they must be born on one of the remaining 364 days. The probability of this occurring is,

Step 3: The Third Individual

A third person enters. They must avoid the birthdays of both previous individuals, reducing the available options to 363 days:

Step 4: The Process Continues

With each additional person, the number of available birthday slots diminishes,

Thus, the probability that all 23 individuals possess unique birthdays is given below,

Step 5: Deriving the Answer

Rather than computing the direct probability of at least one shared birthday, he used the complement principle,

Through computation, estimate,

Thus,

Since this probability exceeds 50% at precisely 23 individuals, he concludes with certainty that 23 is the critical number.

The birthday paradox isn’t just a theoretical concept; it has practical applications in real life, the paradox helps in understand how unlikely events can occur more frequently than we might expect. Here are a few real-world examples of how the birthday paradox is applied in different fields.

Cybersecurity & Password Protection

The birthday paradox plays an important role in cybersecurity, especially in protecting passwords and sensitive data. In security systems, hash functions are used to turn passwords into unique codes that cannot be reversed. However, because of the birthday paradox, two different passwords can sometimes end up with the same hash, which is called a collision. Hackers use this idea in a birthday attack, where they try to create fake data that matches the hash of a real password. To prevent this, companies use stronger encryption methods like SHA-256, which makes it much harder for hackers to guess the right password.

Speeding Up Internet & Cloud Storage

Web browsers and apps store temporary data in a system called cache to make websites and apps load faster. Since there’s limited space, different pieces of data sometimes get stored in the same location, leading to overwriting errors. The birthday paradox helps engineers estimate how often this might happen, so they can design better storage algorithms that reduce conflicts and keep web pages and apps running smoothly. This is especially important for companies like Google, Amazon, and Netflix, which handle large amounts of data every second.

Improving Radar & Sonar Detection

Radars and sonar systems are used in airplanes, ships, and submarines to detect objects and obstacles. These systems send out signals and listen for echoes that bounce back. However, if two different objects return similar signals, the system might mistake one for the other, leading to false detections. The birthday paradox helps engineers predict how often such mix-ups might occur and design better signal-processing systems to make radars more accurate. This is especially important in military defense, air traffic control, and self-driving cars.

Fairness in Lotteries & Raffle Draws

When people participate in lotteries, raffles, or lucky draws, the goal is for each person to have a unique chance of winning. However, if a large number of people pick the same numbers, it increases the chances of multiple winners. The birthday paradox helps lottery organizers estimate how often this happens, allowing them to adjust rules to make sure the game remains fair. This principle is also used in casinos and gambling systems to ensure that randomness is maintained and games cannot be easily manipulated.

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