Questions tagged [birthday]

Birthday problems typically look at probabilities and expectations of a random group of individuals sharing birthdays and how this changes as the number of people increase. They often assume that individuals' birthdays are independently uniformly distributed across 365 days but similar problems can use other numbers or assumptions. They can be generalised to wider occupancy and collision problems.

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1answer
375 views

Is this like the Birthday Problem? Poisson Halloween Party

Suppose that there are n guests at a Halloween party, and that each is wearing one of 200 possible costumes available at local store, uniformly at random and independently of all other guests. Using ...
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Find $n$ such that $365 \left(1-\left(\frac{364}{365}\right)^n - n\frac{364^{n-1}}{365^n} \right) > 1$

I need to find $n$ such that $365 \left(1-\left(\frac{364}{365}\right)^n - n\frac{364^{n-1}}{365^n} \right) > 1$. The answer is $n \ge 28$. Expanding the equation out, rearranging it, taking ...
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Birthday Paradox from different perspectives

When there’s 23 people in a group, the chances that 2 or more people have the same birthday is: $1-\bigl(\frac{364}{365}\bigr)^{253}\approx0.5005$ which is found by taking the chances that a pair ...
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Estimate the probability that N random 4-digit pin numbers are all distinct for N=10, 1000, 100

As per title, the question is "estimate the probability that N random 4-digit pin numbers are all distinct for N=10, 1000, 100". My current working is as follows: There are $10^{4}$ total 0-10 4 ...
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1answer
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Birthday paradox question

I am solving the question: How large must a class be to make the probability of finding two people with the same birthday at least 50%? The first solution I came up with is rather simple. It's based ...
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2answers
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Probability that $n$ persons share their birthdays on exactly/at most $k < n$ months?

I have two exercises. What is the probability that $n$ independent persons share their birthdays on exactly $k<n$ months? What is the probability that $n$ independent persons share their ...
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Expected number of hash insertions until next collision

Other questions like this start with an empty hash, and look for the expected number of hash insertions until the probability of a collision is 0.5. I'm starting off with a hash with existing keys, ...
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Birthday Problem: Expected Number of Birthdays

For a group of $130$ people, assuming that each person is equally likely to have a birthday on each of $365$ days in the year, compute a) the expected number of days of the year that are birthdays of ...
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Birthday Paradoxon

So I saw this interesting problem: https://en.wikipedia.org/wiki/Birthday_problem And I am not the best at probability, so my question is why I cant calculate the probability with P (2 in n same ...
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Probability Two People are Born on the Same Date (alternative way)

What is the probability two people (individuals) will have the same exact birthday? There are 365 days in a year and I assume that any person can be born on any random day, so uniformly. I like to ...
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1answer
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Trying to solve the card shuffle and birthday paradox question

I'm trying to solve the question as to whether two 52 card shuffles have ever been the same but I can't calculate the final step. Additionally I can't find any answer on the internet that has ...
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Probability that N people have the same birthday [duplicate]

This question is from the book "Heard on the Street." There are N = 25 people at a party. One person asks everybody to announce their birthdays and for anyone who has the same birthday as someone ...
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What is the probability of 1st January being the birthday of two persons among 500?

$\mathbf{Question:}$ What is the probability of $1$st January being the birthday of $two$ people when chosen (two persons are picked randomly) among $500$ people? ($365$ days in a Year). $\mathbf{...
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Calculate the probability that no two of three people have the same birthday

Hello I would like for your guidance to the following problem: There are 3 people. We would like to calculate the probability that no two of these people have the same birthday. The probability is $(...
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1answer
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400 people are in a room. What is the probability of two random people having the same birthday? [duplicate]

There are 400 people in a room. I pick two people at random. What is the probability that they have the same birthday? I know that there must be two people in the room who share the same birthday ...
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What is the probability that in a company of $500$ people, only two persons will have birthdays on New Year’s Day?

What is the probability that in a company of 500 people, only two persons will have birthdays on New Year’s Day? I feel the answer is =$\frac{1}{365}\cdot\frac{1}{365}\cdot\frac{364}{365}\cdot\frac{...
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1answer
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Poisson paradigm in “near-birthday problem” example

I am presented with an example called the "near-birthday problem": What if we want to find the number of people required in order to have a 50-50 chance that two people would have birthdays within ...
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1answer
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What is the expected number of days of the year that are a birthday of at least one of n people?

I am new to probability and am just learning about the birthday problem. There is a sample problem that asks among a group of $n$ unrelated people, what is the expected number of days of the year that ...
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1answer
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Is there a generalized solution to the birthday problem? [duplicate]

The problem of the calculating the probability that there is a birthday shared by at least 2 people in a group of size n is well known. I am wondering if there is a way of finding the probability of ...
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Help with birthday problem

I am trying to figure out the probability of having $3$ or more birthdays (out of $23$ people) within the span of $7$ days. I was thinking of going with the number of weeks in a year ($52$), but I ...
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Probability no male- female pairs share same birthday

There are 8 people in a room. There are 4 males(M) and 4 females(F). What is the probability that there are no M-F pairs that have the same birthday ? It is OK for males to share a birthday and for ...
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Birthday problem-Probability exactly $2$ triples and $4$ pairs if $20$ people in room

Say there are 20 people in a room. What is the probability there are exactly 2 triples and 4 pairs. Is my answer shown below correct? Assume 365 days in the year. $P= \dfrac{\binom{365}{2}\binom{363}{...
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1answer
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Probability problem, a variant of the birthday problem

There are $m$ bins and $n$ identical balls. We randomly (uniformly) throw the balls to the bins. Then we count the number of bins which only have 1 ball in it. What is the probability that this number ...
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There are 3 people in a room.What is the probability atleast 2 of them share same birthday

I want to know how to solve by NAIVE APPROACH, NOT by taking complement. That is, not by calculating the probability that no one shares a birthday and subtracting from 1 to get the answer.
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Two solutions to two people having the same birthday

I came across the birthday problem on a forum and I found that there were two answers. My knowledge of probability is pretty limited so I can't quite understand the difference. X and Y having the ...
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Variation on Birthday Paradox Problem

So, I have read the birthday paradox problem, and now I came across below question: Assuming the following: there are no leap years, all years have $n = 365$ days and that people’s birthdays are ...
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Birthday problem: By symmetry, the variance is $\text{Var}(X) = 365 \text{Var}(I_1) + 2 {365 \choose 2} \text{Cov}(I_1, I_2)$.

I have the following problem: Let $X$ be the number of distinct birthdays in a group of 110 people (i.e., the number of days in a year such that at least one person in the group has that birthday). ...
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Birthday problem: $X = \sum_{j = 1}^{365}$ with $I_j \sim \text{Bern}(p)$ where $p = 1−(364/365)^{110}$

I have the following problem: Let $X$ be the number of distinct birthdays in a group of 110 people (i.e., the number of days in a year such that at least one person in the group has that birthday). ...
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1answer
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Birthday paradox with a twist

My questin is a variation on the birthday paradox problem: The difference being that here we want to know if two people have the same given birthday, not any same birthday. How would I solve this?
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Is there a way to *directly* calculate the probability that two or more people (of four chosen) have the same birthday?

You randomly choose four people. What is the probability that two or more of them have the same birthday? It's easy to calculate the probability that none of them have the same birthday (($365*364*363*...
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3answers
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If people sharing the same birthday raise their hand, how many hands do you expect to see raised?

The following question is taken from an interview book assuming that no calculator is provided. Question: There are $25$ people at a party. One person asks everybody to annouycne their birthday, ...
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1answer
108 views

Probability that two people share the same birthday?

Suppose a room contains $n$ people. What is the probability that at least two people share the same birthday? Let $A$ be the probability that at least two people have the same birthday. I know that ...
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1answer
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Find the probabilities for the various configurations of the birthdays of $30$ people born in a leap year?

Find the probabilities for the various configurations of the birthdays of $30$ people born in a leap year. My guess is that total configurations are $\binom{366}{30}$. Is that right? These are for ...
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Probability Jury Problem [closed]

A bailiff who works in the court system asks 12-person juries when they are first sequestered, to break the ice, to play the following game. The jurors sit around a circular table that could be ...
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Probability of distribution using a discrete random function

Problem: Given P persons (say 1000) distribued among N rooms (say 50) using a discrete random distribution function. What would be the probability of a room having at least K persons (say 30 or more ...
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Generalized birthday paradox, expected number of birthday days

Suppose we have m objects and we draw one uniformly n times with replacement. Some objects will be drawn at least once, some never. What is the expected value of the number of objects that are drawn, ...
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1answer
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What is the probability that in a group of n people chosen at random, there are at least two born in the same month of the year?

So I'm working on a probability problem: In Exercise 19 assume it is equally likely that a person is born in any given month of the year. b) What is the probability that in a group of $n$ ...
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Same birthday for two people with and without replacement

It is known that for choosing 2 people out of n with birthdays A,B the probability that A=B is $\frac{1}{365}$. In this case we have replacement which means that we may choose the same person twice. A ...
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Probability 33 people in a group of 100,000 have the same birthday? Assuming years have 365,000 days.

Probability 33 people in a group of 100,000 have the same birthday? Assuming years have 365,000 days. Would this Poisson Formula work? Updated! Note to OP: Probability of exactly $33$ people have ...
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how many persons need to be in one room , so AT LEAST 3 people have the same Birth MONTH?

how many persons need to be in one room , so AT LEAST 3 people have the same Birth MONTH ? Another Answer i suggest : (correct me if i'm wrong) there should be 15 People at least , because if there's ...
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What is the probability that two people's birthdays are always on the same day of the week?

This is related to the Doomsday rule. Certain days of the year, such as 4/4 and 6/6 are always on the same day of the week as each other in a given year. A first approximation would be $\frac17$. ...
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Probability of match in three elements choosing from a group

One of my teachers asked all 26 ($t$) of the students in our class to randomly choose 5 ($k$) exercises from a website from a set of 20 ($n$). He then said that no two students in the class should ...
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1answer
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Repeated trials of the birthday paradox, plus

If: There are 32,768 possible dates in a year (instead of 365) and people have a completely uniform random distribution of birthdays. In each room there will be 3 people (if easy enough, alternate 3, ...
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Birthday paradox for large numbers

I'm attempting to test the claim: "Every card deck shuffled is unique. A shuffled deck of cards will exist once and never again." Assumption: A perfect world where a deck of cards is perfectly ...
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Distribution for count of unique elements

Consider the selection, with replacement, of $N$ elements uniformly at random from $\{1, 2,\dots, M\}$. From the selected elements, count the number of unique elements. Does the distribution of this ...
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1answer
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What do Gegenbauer polynomials have to do with the birthday problem?

I was reading up on Gegenbauer polynomials for a blog post about the kissing number problem, which I also asked a separate question about later. On the Wolfram MathWorld plot summary for Gegenbauer ...
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$100$ persons think of a number between $1$ and $4900$. Probabilty: Thinking of same number vs. $1$ number chosen by multiple people

$100$ persons think independently of a number between $1$ and $4900$. What is more likely? All people have thought of different numbers $(1)$ A number has been chosen by more persons $(2)$ It ...
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1answer
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Birthday problem vs having N known non-duplicates and choosing one more

I understand the birthday problem from a basic level, but I'm curious how checking for a pair after every choice of N alters the probabilities. For example, assuming you want to choose N random ...
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307 views

Probability of at least $3$ people sharing the same birthday in a group of $n$ people

Disclaimer: I've seen posts with good answers for the case of "at least $2$ people", or "exactly $2$ people". Posts with "at least $k$ people" usually suppose that the mutual birthday is a fixed day (...
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Show that $p_{n} \geq 1- \exp{(-n(n-1)/730)}$

On the issue of the birthday paradox,Let $p_{n}$ be the probability that in a class of $n$ at least $2$ have a their birthdays on the same day (exclude $29$ Feb). Use the inequality $1-x \leq e^{-x}$ ...

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