rcor.ro

2022-12-02

## The Problem

• The birthday paradox, also known as the birthday problem, states that in a random group of 23 people, there is about a 50 percent chance that two people have the same birthday.
• There are multiple reasons why this seems like a paradox. New sources will be added it to this ever growing presentation.
1. Scientific American proposes the same problem, creating another paradox, by stating:
1. Every one of the 253 combinations (of two persons) has the same odds, $p=0.99726027$, of not being a match, they say.
2. If you calculate (364/365)^253, you’ll find there’s a 49.952 percent chance that all 253 comparisons contain no matches, they say.

## 3 people only

• Why go for 23, before understanding the problem with only 3 persons?
• Define $A_i$ the event where the $i$ person will not share the same birthday with nobody and the negation by $\bar{A}_i$.
• We have 3 combinations (of two persons) $A_1A_2$, $A_1A_3$, $A_2A_3$.
• $P(A_1A_2)$ denotes the probability of both $A_1$ and $A_2$ being TRUE.
• If $A_1A_2$ and $A_1A_3$ both not being a match (TRUE), then $A_2A_3$ can be FALSE (share the same birthday).
• BUT: $P(A_1A_2A_3|G_3)=P(A_2A_3|G_3)$, so problem solved (one combination), why asking for $p^3$?

## Bayes settings

• We start by declaring the sample space, $G_3$, being a group of 3 persons, and here the negation $\bar{A}_i$ depends on $G_3$.
• The Problem is symmetric, so finding $P({A}_3|G_3)$ will determine $P({A}_2|G_3)$ as well.
• $P(\bar{A}_1\bar{A}_2\bar{A}_3|G_3)=1/365^2$ and this can be generalized for any group
• $P(\bar{A}_1\bar{A}_2\bar{A}_3|G_3)=P(\bar{A}_1|\bar{A}_2\bar{A}_3G_3)·P(\bar{A}_2\bar{A}_3|G_3)\Rightarrow P(\bar{A}_2\bar{A}_3|G_3)=1/365$ (1)
• $P(A_1A_2A_3|G_3)=P(A_1|A_2A_3G_3)·P(A_2A_3|G_3)=P(A_2A_3|G_3)$, by simple logic: if $A_3$ is TRUE and $A_2$ is TRUE, then the first person can’t share the same birthday with nobody else in $G_3$, so the first probability is one. Also holds to be true for any $n>2$: $P(A_1A_2...A_n|G_n)=P(A_1|A_2\dots A_nG_n)·P(A_2...A_n|G_n)$

## Bayes calculations

• (2): $P(\bar{A}_2|\bar{A}_3G_3)=1/2$, by knowing that the third person shares the birthday, we have a 50% chance that will mach either of the remaining persons.
• (1)+(2): $P(\bar{A}_2\bar{A}_3|G_3)=P(\bar{A}_2|\bar{A}_3G_3)·P(\bar{A}_3|G_3)$ so we get $P(\bar{A}_3|G_3)=2/365$ or $P({A}_3|G_3)=363/365$
• Actually $P({A}_2|{A}_3G_3)=364/365$ so only by knowing that the third person is alone, will impose that the other persons will have that claimed chance of being alone (not matching the remaining person, or a $G_2$ problem)
• Finally, $P({A}_2{A}_3|G_3))=P({A}_2|{A}_3G_3)P({A}_3|G_3)=363·364/365^2$

## Another solution

• Even though not recommended in general, Bayes being preferred, sampling all possible different birthdays of 23 persons can pave the way.
• The number of ordered arrangements of 23 days taken from 365 unlike days is $365!/(365-23)!$
• Not to confuse with the number of ways of selecting 23 different days! Not looking for $365\choose23$, because
• We consider the total number of cases $365^{23}$, so we get $P(A_1A_2...A_{23}|G_{23})=364·363\cdots343/365^{22}\approx0.4927$

## Mistakes explained

• $p=P(A_1A_2)$ has different meanings for different groups of people, so claiming that $p=364/365$ it is false in general, but TRUE in $G_2$
• Same mistakes can be found almost everywhere, the difference in calculations are luckily close (49.952% vs. 49.27%)
• Can not mix results from $G_2$ to $G_n$, for $n>2$
• We might guess that the intention was to calculate the probabilities of having 2 different days as a pair and impose it to all pairs. But $({A}_2, {A}_3)$ or any other pair of events are not independent!
• Now imagine there were 366 persons in the group. It is clear that $P(A_1A_2...A_{366}|G_{366})=0$, but $(364/365)^{66795}>0$

## Remarks

• Google returned the selected papers, no judgmental sampling was intended. The author has not been involved in a dispute with the editors of the mentioned papers

• The views expressed in this article are those of the author.

• This is an updated report using the latest information and tries to adapt the main idea using more information as they appear,

• actively pursuing error-correction by creating criticisms of both existing ideas and new proposals.

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