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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What is the probability that three living people in the same family will celebrate their birthdays on exactly the same day.

I celebrate my birthday on the same day as one of my grandchildren. Just wonder how rare it would be for three people in the same family to celebrate their birthdays on the same day.
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1 vote
1 answer
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Progressive Birthday problem

The Birthday Problem is very interesting to me. The more dates you fill up, the lower the chances a date will be outside what has already been seen. But I do not seem to understand at all how ...
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1 vote
1 answer
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probability of 2 person has the same birthday in a class.

problem: there are n persons in a room, what is the probability that no two of them celebrate the same birthday in a year? Here is my thought process, The sample space is $|\{(b_1,b_2,\dots,b_n): b_1,...
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Probability of no birthdays for X consecutive days within a group of n people

Consider a group of n people. Assume 365 days in a year and that birthdays are independent and uniformly distributed. What is the probably that ...
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2 votes
1 answer
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What is the expectation value of the minimum 'distance' between two random 64-bit numbers out of a set of N?

Assume we have a set of N random integer numbers in the interval [0, 2^64> (or equivalent, consider N randomly chosen corners (vectors) of a 64-dimensional ...
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1 vote
1 answer
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At least how many residents of the country have the same birthday?

I am trying to solve a pigeonhole question in discrete mathematics. Let's suppose that a country has 11.000.000 people. At least how many residents of the country have the same birthday? Take into ...
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What is the probability of at least one pair of people who share a birthday and whose mothers share a birthday?

Problem $71$ of Chapter 4 from Introduction to Probability by J. Blitzstein and J. Hwang. In a group of $90$ kids, what is the approximate probability of there being at least one pair of kids born on ...
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1 vote
1 answer
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Which would be the "direct" formula for the birthday paradox , withouth subtracting to 1?

I stumbled upon the birthday paradox, and I get it. However, all the explanations I see solve the probability by subtracting to 1 the probability of all people having different birthdays. What I am ...
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2 votes
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Average number of birthdays today

On Facebook, I can see which friends have birthdays today. Sometimes there are 1, sometimes more than 1, and sometimes zero friends. What's the average number of birthdays today? To formalize: Problem ...
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1 vote
2 answers
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Find the probability that at least two people out of $k$ people will have the same birthday

The Birthday problem. Find the probability that at least two people out of $(k=5)$ people will have the same birthday. The usual approach would be to use $$p=1-\frac{P_{365,5}}{365^5}$$ However, I ...
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Probability distribution for birthday paradox

In Wikipedia we read In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair with the same birthday. If we consider the probability ...
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4 votes
1 answer
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Random Variable - Birthday Problem

How many people are needed so that the probability that at least two of them were born on the same day of the week is at least 1/2? (Assume that the days of the week are equally likely to be the ...
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4 votes
1 answer
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A man invited five friends.

A man invited five friends. He was born in April as also all the invited friends. What is the probability that none of the friends was born on the same day of the month as the host? The way I ...
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0 answers
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Expected number of days for email propagation [duplicate]

Assume you have an email account with 50 friends. Each day, you randomly and uniformly select a friend to send an email to. Meaning that one particular friend can be selected on multiple occasions. ...
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3 votes
2 answers
229 views

Birthday problem - expected number of shared birthdays

Given $m$ people and an $n$ possible "days of the year", what is the expected number of days which 2 or more people share as a birthday (if the distribution of birthdays is iid uniform over ...
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1 vote
1 answer
63 views

$1/2 = 1 - e^{-n(n-1)/2d}$ General Birthday Formula? [closed]

When solving $\frac{1}{2} = 1 - e^{-n(n-1)/2d}$ for $n$, the solution should be $n \approx \sqrt{\ln(4)d}$. But I can't get the calculations done. Any advice would be highly appreciated. Thank you! ...
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Birthday Problem as people enter in the room

On the wikipedia page regarding the birthday problem there are several variations to it: https://en.wikipedia.org/wiki/Birthday_problem One of the variations is this one: as people enter a room one at ...
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3 votes
1 answer
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Solving Birthday Paradox with Triangular Number formula

I have an incorrect solution to the classic birthday paradox question, and while plugging in some values shows the formula is wrong, I can't see any intuitive reasoning as to why it is incorrect, or ...
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How do you calculate the probability of *only* 2 people having different birthdays (excluding leap years)?

I know that the numerical answer is $\frac{364}{365}$, but I didn't understand why. this answer seems to contradict my current understanding of probability. For example, to calculate the probability ...
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2 votes
1 answer
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Asymptotics of expected number of draws until repeat

Suppose there are $n$ distinct balls in a bag and they are drawn with replacement until the first repeat. Let $X$ be the number of balls drawn. I have shown that the distribution is unimodal and that ...
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Why is the birthday problem not so surprising?

I'm reading Blitzstein/Hwang's "Introduction to Probability". Here, they give an explanation as to why the conclusion obtained in the birthday problem is not so surprising: I don't quite ...
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How to bound chances of edges from a set in a random matching?

suppose I have n nodes, and I pick at random a partial matching (i.e. disjoints edges) containing $$\alpha n$$ edges. How can I show the probability of having at least one edge which both of its ...
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2 answers
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Solve inequality involving product of x terms [closed]

What is the minimum value of $x$ such that $$1-\prod_{k=0}^{x-1}\left(1-\frac{k}{2^{160}}\right)>0.01$$ is true? Is there any easy way to solve this without using computational software?
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2 votes
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Probability of at least 3 people sharing a common birthday

What is the probability that 3 or more people share a common birthday, in a group of 160 people? Approach: We have: $P(X\geq 3)= 1-[P(X=0)+P(X=2)]$. (where $X\geq i$ means at least $i$ people share a ...
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Birthday paradox problem in terms of months

Question - Consider the following- It is equi-probable to have a birthday in any month of the year. 6 random people are put into a room. Q1. Find the probability that at least two people have a ...
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What is the probability of two random elements from two lists are equal? (formal proof)

Suppose $L_1$ and $L_2$ are the lists of random uniform elements with sizes $m$. The elements are from $\{0,1\}^{n}$, e.g. the binary vector in $F_2^n$. What is the probability that $x_1 \in L_1$ and $...
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1 vote
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Birthday problem but with $2^{128}$ different days in the year [duplicate]

I am trying to calculate how many randomly generated ids I need to produce for there to be a 1% probability I get a duplicate id. There are $2^{128}$ possible ids. I understand this is just the ...
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1 vote
1 answer
115 views

What is the probability that with 100 people, there are 3 consecutive birthdays at some point in the year?

Assumptions: $365$ days in the year. All birthdays are equally likely. The three consecutive days must fall within the same calendar year. This was a fun problem that came up at my office of $100$ ...
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1 vote
2 answers
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Expected number of aces in a poker hand

So I'm having difficulty understanding a certain idea. Say I have a standard $52$ card deck (fairly shuffled, etc.) and I pull $5$ cards from it. We ask, what is the expected value of the number of ...
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2 answers
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Using R, solve the birthday paradox

The probability that two students in a class have the same birthday is at least 75%. What is the minimum size of the class? I tried, ...
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1 vote
3 answers
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Probability of 4 parents sharing 2 birthdays [closed]

I'm a math idiot and I'm trying to figure something out. My mother was born on April 10th as was my friend's mother. My father was born on July 4th as was my friend's father. How do I calculate the ...
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Probability- Birthday problem

The traditional way of finding the answer is to find the complement and subtract from one. I tried an alternative solution. I come up with a formula. for $k$ people, $P(k)=P(k-1)+{(k-1)\over365}\...
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-3 votes
1 answer
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Probability of birthday [closed]

In this years IMO a decision has been taken that each team will be consist of 10 members. What is the probability that at least two people will have a birthday on the same day of the week?
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1 vote
1 answer
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At least 2 share the same birthday but without doing actual calculations for the probability

I am reading the following problem: If $20$ people are selected at random, find the probability that at least $2$ of them have the same birthday. As a follow up, how large a group is needed to give a ...
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closed form of multiple in the probability of a two to one collision problem (birthday paradox).

I have a problem where I need to find the chance of finding a collision in a two to one set of size $N$. For example: Let's say I have 10 indices. There are then 5 pairs that map to the same output. ...
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1 vote
0 answers
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Birthday problem for people knowing each other

Suppose a large chat room for a country, what is the probability that two people know each other in real-life? We have: n: The number of people in the chat room d: The number of persons the average ...
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1 vote
1 answer
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Show that in a group of four people it is more likely than not that two of them were born on the same day of the week. [closed]

I've started the working : Possible combinations : AB AC AD BC BD CD = 6 P(2 people born same day of week) = 1 - P(No people born on same day of week) =1−((7/7)⋅(6/7)⋅(5/7)⋅(4/7))=1−(120/343)=223/343≈...
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0 votes
1 answer
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Unclear how to prove this logarithmic reduction of a sequence, in reference to the birthday problem

On page 46 of Rouse Ball and HSM Coxeter's Mathematical Recreations and Essays, 13th ed., appears this step, in reference to the birthday problem $$ (1-\frac{1}{365})(1-\frac{2}{365})...(1-\frac{n-1}{...
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10 votes
3 answers
884 views

Birthday Paradox with Leap Year

I looked online, and found more than one and inconsistent answers to the Birthday Paradox when we throw the leap year into the mix. None of the answers I saw match with my own. I am posting my ...
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-2 votes
2 answers
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Birthday Problem: Expected number of people in a room

If an infinite amount of people enter a room one by one, what is the expected number of people in the room when you first find two that share the same birthday? (Assuming no leap years and every ...
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1 vote
1 answer
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Given a concrete day, probability that two people have birthday on the same day.

What is the probability that on a specific day (e.g. December 24) two people have birthday in a room with 23 people? The probability, given a specific day, one person has birthday is in my opinion $\...
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1 vote
2 answers
175 views

Calculation of probability that at least 2 out of n people have the same birthday leads to paradox

Usually, the way the probability that at least 1 pair out of n people have the same birthday is calculated is by calculating the probability of the converse statement - the probability that no pair ...
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2 votes
2 answers
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Birthday probability question

I came across a variation of the birthday problem asking "in a room of $4$ people what is the probability that at least $3$ of them share the same birthday". I was unsure of the answer and ...
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3 votes
1 answer
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Why is this solution for the birthday problem wrong?

How many people do you need in the same room for the probability $p$ to be at least $0.5$ for two people to have the same birthday? My idea is that for every pair of people the probability that their ...
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5 votes
3 answers
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The Birthday Paradox with combinatorics

Assume a year has $365$ days, how many are required to have a $50%$ chance of $2$ people having the same birthday? According to Scientific American, there are $23$ people needed to achieve the goal. $$...
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2 votes
0 answers
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Birthday Problem: Expected Number of people with common birthday [duplicate]

This might be a different variant of the typical birthday problem. Given a room of $n$ people, let $N$ be a random variable representing the number of people who have a birthday common with at least ...
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4 votes
1 answer
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Birthday problem with large $n, d$ values

In the Birthday problem, the formulas $${\displaystyle {\begin{aligned}p(n;d)&={\begin{cases}1-\displaystyle \prod _{k=1}^{n-1}\left(1-{\frac {k}{d}}\right)&n\leq d\\1&n>d\end{cases}}&...
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  • 1,597
2 votes
1 answer
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My try of the Birthday Paradox

I have seen the solution of the paradox on Wikipedia, but before that I tried it out myself. But I got my answer wrong with my approach. To calculate the probability of no two people in a group of 23 ...
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6 votes
1 answer
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Variation of Birthday problem - Group of n people

I know this has been posted several times and I have gone through most of the relevant posts. Here is one which I am having a difficult time to solve: There are 450 people in a room; (1) how many of ...
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2 votes
1 answer
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Birthday problem (combinatorics), without using inverse solution

There is that classic question about how many people in a room is required so that at least one pair of people share a birthday, with > 50% probability, the answer is 23. The standard textbook ...
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