# 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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### Unknown distribution for birthday problem

Coming from Blitzstein's book: In the birthday problem, we assumed that all 365 days of the year are equally likely (and excluded February 29). In reality, some days are slightly more likely as ...
1 vote
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### Family Members Birthday Dates all different, but our birthdays will fall on same day, even Leap Years. There is a total of 9 in this Birthday Club. .

I can compile a list if needed and post, but I noticed this over 50 years ago, My Father, My Brother and Myself our Birthdays fall on the same day of the week every year. Even Leap years, that does ...
1 vote
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### Birthday Problem: Confusion between PMF and CDF -

The question: (Introduction to Probability, Blitzstein and Nwang, p.128) People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing ...
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### How many people must be in a room until it is at least a $50\%$ chance that two will have the same amount of change?

Book problem: If the amount of change in a pocket is assumed to be uniformly distributed from $0$ to $99$ cents, how many people must be in a room until it is at least a $50\%$ chance that two will ...
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### The Birthday paradox with variable-likelihood birthdays. [duplicate]

I know that the Birthday Paradox is the fact that in a room of 23 people, the chances are more than 50 percent that at least two people share a birthday. However, this is under the assumption that all ...
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### Birthday Problem Confusion Using the Counting Rule

I am stumped by the below confusion: Question: How many people do we need in a class to make the probability that (at least) two people have the same birthday more than 1/2? (For simplicity, assume ...
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### Number of date collisions in birthday problem

If I generate uniform random integers from 1 to K and count how many unique numbers I get $n_\mathrm{unique}$, I empirically obtain: the mean is: $\frac{2K}{\pi}$ the variance is $\frac{K}{\pi^{2}}$. ...
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### How can I prove that the probability that exactly 2 people share the same birthday is more likely than everyone has a different day out of 20 people? [closed]

I will be thankful if you can help me and show how to solve this.
1 vote
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### Miscalculating Probability of At Least $2$ People Having The Same Birthday

Regarding the problem: choosing 23 people randomly, show that there is greater than a $50$ percent chance that at least two of them will have the same birthday. What is the error in the way I'm trying ...
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### Classmate birthday Probability [duplicate]

I dealt with one issue, namely: Consider $k$ independent realizations of a random variable uniformly distributed over a set of $n$ values. 1 What must $k$ be for the probability that the given outcome ...
57 views

### Birthday-esque problem, but for 2 pairs, or a triple

Let's say I've got a pool of 20 numbers, and each event chooses a number randomly. I'm trying to find the 50% point for one of these three: 50% chance that by this event, at least 1 duplicate number ...
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### Help with deriving solution for multiple birthday problem

I've been thinking about one version of the more general birthday problem, namely for the case of k $\ge$ 3. I found this document explaining the solution through a combinatorial method, but I'm ...
1 vote
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### How to work out the probability of two random sequences sharing a certain number of matches?

Pick two sequences of numbers, $S_1$ and $S_2$. $S_1$ is $n_1$ picks from $1$ to $k$, $S_2$ is $n_2$ picks from $1$ to $k$. There could be duplicates within each sequence, for instance $S_1$ might ...
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### Collisions in a Sample [closed]

Based on birthday paradox; Let $d$ be the set of elements randomly chosen from a set of $n$ distinct elements then a) What is expected number of unique elements in $d$ (remaining will be repetition of ...
1 vote
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### Birthday problem: Poisson vs binomial random variable

From this post, the birthday problem involving more than 2 people can be approximated using a Poisson random variable. But I am wondering whether a binomial random variable can be used here. I imagine ...
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### Birthday Paradox at least Vs Exactly

The famous paradox in probability theory, the Birthday Problem asks that:” What is the probability that, in a set of n randomly chosen people, AT LEAST two will share a birthday.” In some other books ...
1 vote
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### Birthday paradox - variance, parallelisation, simple proofs?

Suppose we sample uniformly random elements from a set of cardinality $n$, and save them in a table. We continue doing this process (each sampling is one step) until we get a collision. What is the ...
114 views

### What is the probability of sharing a birthday if a year has an infinite number of days?

Here is the problem: Suppose that there are $k$ people. Each of them independently picks a uniformly random number from the set $\{1, 2,...,n\}$. We say that a collision happens if there exist two ...
67 views

### How many days with birthdays are in a classroom?

Assumption: I am a teacher of a classroom with n students. And every time there is one or more birthdays in a day, I will buy only a cake. Question: How many cakes do I have to buy on average every ...
1 vote
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Let us assume every child in the world has a random and uniform favorite number between $1$ and $m$, and also has a different random and uniform unfavorite number between $1$ and $m$. Denote $E_{k,m}$ ...
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### Birthday problem with really high parameters

I have $n$ objects. Every object has a random value in $[0;k)$ (in $\mathbf{N}$). How high is the probability for every object to be unique within the set of $n$ objects? This is obviously a case of ...
1 vote
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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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### 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 ...
236 views

### 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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### 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 ...
1 vote
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### $1/2 = 1 - e^{-n(n-1)/2d}$ 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! ...
63 views

### 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 ...
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 ...