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Probability Of Two Person Having Same Birthday

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The Surprisingly High Probability of Shared Birthdays



We often encounter probability in everyday life, from predicting the weather to assessing risks. One fascinating and counter-intuitive example is the probability of two people in a group sharing the same birthday. Intuitively, you might guess that you'd need a very large group for this to be likely, but the reality is quite surprising. This article will delve into the mathematics behind this phenomenon and explain why it's much more probable than you might think.

Understanding the Problem: Why Not 1/365?



The most common initial approach is to assume the probability of two people sharing a birthday is simply 1/365 (ignoring leap years for simplicity). This is incorrect. This calculation only considers the probability of one specific person sharing a birthday with another specific person. We're interested in the probability of any two people in a group sharing a birthday. The difference is crucial. We're not looking for a specific match; we're looking for any match within the group.

The Complementary Approach: Focusing on the Opposite



Instead of directly calculating the probability of a shared birthday, it’s much simpler to calculate the probability of no shared birthdays and then subtract that from 1 (representing 100% probability). This is known as using the complement rule.

Let’s consider a small group:

Person 1: Can have any birthday (365 possibilities).
Person 2: To not share a birthday with Person 1, they must have one of the remaining 364 days. The probability of this is 364/365.
Person 3: To not share a birthday with either Person 1 or Person 2, they must have one of the remaining 363 days. The probability is 363/365.

This pattern continues for each additional person. To find the probability of no shared birthdays in a group of 'n' people, we multiply the probabilities together:

(365/365) (364/365) (363/365) … (365-n+1)/365

The probability of at least two people sharing a birthday is then 1 minus this result.

The “Birthday Paradox” in Action: Illustrative Examples



Let's look at a few examples to see how quickly the probability rises:

2 people: The probability of a shared birthday is approximately 0.0027 or 0.27%. This is quite low.
10 people: The probability jumps to about 0.117 or 11.7%. Still relatively low, but significantly higher than with just two people.
23 people: This is the magic number! The probability of at least two people sharing a birthday exceeds 50%. This is the heart of the “birthday paradox”.
50 people: The probability rises to an astounding 97%.

These numbers highlight the non-intuitive nature of the problem. The probability increases dramatically as the group size grows, even though the individual probabilities remain small.

Why It’s More Likely Than You Think



The reason the probability rises so quickly is because we're considering all possible pairings within the group. With a larger group, the number of potential pairs increases dramatically, leading to a higher chance of a match. It's not about the individual chances but the cumulative effect of multiple comparisons.

Key Takeaways and Insights



The “birthday paradox” illustrates a fascinating principle in probability: seemingly unlikely events can become quite probable when considering multiple possibilities simultaneously. It demonstrates that intuitive estimations can often be misleading, and that a rigorous mathematical approach is often necessary for accurate calculations. This principle has implications in various fields, including computer science (hash collisions) and cryptography.

FAQs



1. Does this calculation account for leap years? For simplicity, we've ignored leap years. Including them would slightly alter the probabilities but not significantly change the overall result.

2. What if the birthdays are uniformly distributed? The assumption of uniform distribution of birthdays is generally valid, although slight variations exist in real-world birth data.

3. Why is it called a “paradox”? It's called a paradox because the result is counter-intuitive; we expect a much larger group before the probability of shared birthdays becomes high.

4. What are the real-world applications of this concept? This concept is used in cryptography (hash collisions), analyzing data distributions, and understanding the likelihood of coincidences.

5. Can this be generalized to other events besides birthdays? Yes, the principle applies to any event with a finite number of possibilities, as long as the probabilities are relatively uniform. For example, you could analyze the probability of two people in a group having the same last digit in their phone number.

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Probability that any two people have the same birthday? 12 Oct 2019 · In Blitzstein's Introduction to Probability, it is stated that the probability that any two people have the same birthday is 1/365. However, isn't this the conditional probability that the second person has the same birthday as the first person, given the birthday of the first person?

The Birthday Problem - University College Dublin By the pigeonhole principle, you would need to have 366 people in a room in order to have a 100% chance (a guarantee) that at least 2 people share the same birthday (Note: for this workshop, we are assuming a 365-day year. However, if using the leap year model, just add one to the number of days). Q1.

Probability that two people have the same birthday in a party? 19 Jun 2017 · Let's find the probability that no two people have the same birthday. We have $\displaystyle \frac{365}{365} \times \frac{364}{365}...\frac{365-n+1}{365}$ Because each person must have a new birthday.

What is the probability that in a group of two people, both will have ... 29 Jul 2021 · Probability of occurring = 1 - the probability of not occurring. Let’s calculate for the probability of not occurring, i.e. probability such that both of them don’t have a birthday on the same day. For suppose the first person has a birthday on a particular day then the other person can have a birthday in the remaining 364 days.

Probability and the Birthday Paradox | Scientific American 29 Mar 2012 · In this case, if you survey a random group of just 23 people there is actually about a 50–50 chance that two of them will have the same birthday. This is known as the birthday paradox. Don't...

Birthday problem - Wikipedia In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday. The birthday paradox refers to the counterintuitive fact that only 23 people are needed for that probability to exceed 50%.

Probability of two people having the same birthday Denote the probability of a given person's birthday being on the k k th day of a year of n n days by pk p k. Then, the probability that two random people have the same birthday is. P =∑k=1n p2 k. P = ∑ k = 1 n p k 2. In particular, when p1 = ⋯ =pn p 1 = ⋯ = p n, this probability is 1 n 1 n.

Birthday Paradox Calculator Explore the intriguing Birthday Paradox with our easy-to-use calculator. Discover the probability of shared birthdays in a group and dive deep into this statistical phenomenon.

How to calculate the chance of two persons sharing the same birthday Assuming that birthdays are equally likely to happen on each of the 365 365 days of the year, the probability, P(2) P (2), that Person 2 2 has a different birthday than Person 1 1 is. This is because, if Person 2 2 was born on any of the other 364 364 days of the year, Persons 1 1 and 2 2 will not share the same birthday.

Birthday Paradox - GeeksforGeeks 19 May 2023 · What is the probability that two persons among n have same birthday? Let the probability that two people in a room with n have same birthday be P(same). P(Same) can be easily evaluated in terms of P(different) where P(different) is the probability that all of them have different birthday.

How does the birthday paradox work? — Today You Should Know 30 Jan 2025 · 🎂 The birthday paradox: Find the probability of no matches at all. Returning to the birthday paradox, it states that there's an over 50% chance of a group of 23 people having at least two people sharing the same birthday. First we need to find the probability of no one in that group of 23 people sharing a birthday. The formula for that is:

Birthday Paradox Calculator The birthday paradox calculator allows you to determine the probability of at least two people in a group sharing a birthday. All you need to do is provide the size of the group. Imagine going to a party with 23 friends. What is the probability that at least two …

What is the probability of exactly two out of n persons sharing a birthday? The classical Birthday problem asks for the probability of at least two out of $n$ people sharing the same birthday or sometimes for the least amount $n$ of people required such that with a probability of 50% at least two of them share their birthday.

Same Birthday Odds: The Birthday Paradox - Statistics How To It stands to reason that same birthday odds for one person meeting another are 1/365 (365 days in the year and your birthday is on one of them). But it may surprise you to know that with a class of 23 people, there is a 50% chance that two people will share a birthday.

The Birthday Paradox: How Likely Is It for Two People to Share the Same ... 19 Oct 2024 · It turns out that in a room of just 23 people, there’s about a 50% chance that two of them share the same birthday! That probability skyrockets to 99.9% with 70 people. How does this work,...

Two people were born in 2024 what is the probability of them having same 10 Feb 2025 · To find the probability that two people have the same birthday, we can use the concept of probability in a uniform distribution. Assuming there are 365 days in a year (ignoring leap years), the probability that the second person has the same birthday as the first person is given by the formula: P (s am e bi r t h d a y) = 365 1 .This is because there is only one day …

Birthday Problem Paradox Calculator - Online Probability - dCode.fr Tool to calculate the birthday paradox problem in probabilities. How many people are necessary to have a 50% chance that 2 of them share the same birthday.

The Birthday Paradox: Why a Room of Only 23 Strangers Has a … 7 Jan 2025 · If you have 23 people in the room, you have 253 birthday pairings. For the first pairing, there’s a 1 in 365 odd (0.27%) that they have the same birthday, and a 364 in 365 odd (99.73%) that they ...

Probability of Shared Birthdays - BrownMath.com 25 May 2003 · (a) What is the probability that any two people have different birthdays? The first person could have any birthday (p = 365÷365 = 1), and the second person could then have any of the other 364 birthdays (p = 364÷365).

Understanding the Birthday Paradox – BetterExplained The chance of 2 people having different birthdays is: Makes sense, right? When comparing one person's birthday to another, in 364 out of 365 scenarios they won't match.

Calculating the Probability of Two People Having the Same Birthday To calculate the probability that two people selected at random have the same birthday, we need to consider the number of possible outcomes and the number of favorable outcomes. First, let’s determine the number of possible outcomes.

3.4: Probability and Compound Events - Statistics LibreTexts 29 Jan 2025 · We could have the first toss produce a \(1\) while the other two do not, or we can have the second produce a \(1\) and the other two not, or we can have the first two tosses both produce a \(1\) and the third not, or many other possibilities. We must determine many probabilities to use our rule with "or." However, there is an easier way.

probability - At least two people have the same birthday 13 Jan 2018 · You can read all about this famous problem here to learn how to calculate the probability that at least two of $n$ people share a birthday. In your case at least two of $85$ people will share a birthday with a probability of approximately $99.998\%$.

The Birthday Paradox Explained - Built In 1 Aug 2024 · The birthday paradox is a probability theory that states that the probability of two people in a group sharing the same birthday grows based on the number of pairings, not the number of people in a group.