Posts: 864
Threads: 84
Joined: July 26, 2017
Reputation:
5
What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 2:06 am
The probability that 2 out of 23 people will share the same birthday is slightly higher than 50%, and that is the famous Birthday Paradox. However, what is the probability that 3 out of 23 people will? I have, like I have written in this article about something distantly related to that, estimated numerically using Monte Carlo method that it is around 1.26%, and that the probability that 4 out of 23 people will share the same birthday is around 0.018%. However, I am interested whether there is a general formula for that.
Posts: 20443
Threads: 447
Joined: June 16, 2014
Reputation:
111
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 2:33 am
(This post was last modified: January 21, 2022 at 2:34 am by ignoramus.)
ssssssss
No God, No fear.
Know God, Know fear.
Posts: 864
Threads: 84
Joined: July 26, 2017
Reputation:
5
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 4:57 am
(January 21, 2022 at 2:33 am)ignoramus Wrote: ssssssss
What does that mean?
Posts: 35183
Threads: 492
Joined: July 24, 2013
Reputation:
109
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 5:09 am
It's one of those paradoxes that isn't. Here's the roughest way to look at it.
If you have a group of 23 people, the number of possible pairs is 253: 23x22/2. The odds of shared birthdays is found by dividing the possible pairs by the number of days in a year. 253/365 is just over 69% (the actual percentage is much closer to  but still above  50%).
Boru
‘Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.’ Ralph Waldo Emerson
Posts: 20443
Threads: 447
Joined: June 16, 2014
Reputation:
111
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 5:26 am
(January 21, 2022 at 4:57 am)FlatAssembler Wrote: (January 21, 2022 at 2:33 am)ignoramus Wrote: ssssssss
What does that mean?
No God, No fear.
Know God, Know fear.
Posts: 864
Threads: 84
Joined: July 26, 2017
Reputation:
5
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 5:44 am
(January 21, 2022 at 5:09 am)BrianSoddingBoru4 Wrote: It's one of those paradoxes that isn't. Here's the roughest way to look at it.
If you have a group of 23 people, the number of possible pairs is 253: 23x22/2. The odds of shared birthdays is found by dividing the possible pairs by the number of days in a year. 253/365 is just over 69% (the actual percentage is much closer to  but still above  50%).
Boru
Right, it is more like the Monty Hall Problem: something which is not hard to analyze mathematically, but the result is surprising to most people.
Posts: 5141
Threads: 446
Joined: December 12, 2015
Reputation:
23
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 6:32 am
(This post was last modified: January 21, 2022 at 6:45 am by Jehanne.)
Yes, there is a general formula, which can be reasoned as follows:
1) The probability of an event plus its complement (the event not occurring) is 1 (or, "unity").
2) While not being entirely true, we may assume, for convenience, that every day of the year is equally likely to be born on.
3) The probability of being born on any one day is 1/365 for nonleap years.
4) If a single person is born on one day, there are 364 other days for another individual to be born on, such that the two individuals do not have the same birthday. For the third individual, there would be 363 days, etc.
5) 1  (365 * 364 * 363...) / 365 ^ n would give the probability that 2 or more individuals would have the same birthday.
P.S. Okay, it's 4 AM, and, so, I misread your question a bit. Yes, getting the probability for exactly three individuals is quite a bit more tricky.
And without delay Peter went quickly out of the synagogue (assembly) and went unto the house of Marcellus, where Simon lodged: and much people followed him...And Peter turned unto the people that followed him and said: Ye shall now see a great and marvellous wonder. And Peter seeing a great dog bound with a strong chain, went to him and loosed him, and when he was loosed the dog received a man's voice and said unto Peter: What dost thou bid me to do, thou servant of the unspeakable and living God? Peter said unto him: Go in and say unto Simon in the midst of his company: Peter saith unto thee, Come forth abroad, for thy sake am I come to Rome, thou wicked one and deceiver of simple souls. And immediately the dog ran and entered in, and rushed into the midst of them that were with Simon, and lifted up his forefeet and in a loud voice said: Thou Simon, Peter the servant of Christ who standeth at the door saith unto thee: Come forth abroad, for thy sake am I come to Rome, thou most wicked one and deceiver of simple souls. And when Simon heard it, and beheld the incredible sight, he lost the words wherewith he was deceiving them that stood by, and all of them were amazed. (The Acts of Peter, 9)
Posts: 24977
Threads: 513
Joined: June 16, 2015
Reputation:
90
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 8:49 am
I don't share my birthday with anybody, the cake is all mine damnit.
I don't have an anger problem, I have an idiot problem
Posts: 9488
Threads: 66
Joined: July 10, 2013
Reputation:
61
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 10:04 am
Another fascinating thread.
“Two things are infinite: the universe and human stupidity; and I’m not sure about the universe.” ~Albert Einstein
Posts: 864
Threads: 84
Joined: July 26, 2017
Reputation:
5
RE: What's the probability that 3 out of 23 people will share the same birthday?
January 21, 2022 at 11:33 am
(This post was last modified: January 21, 2022 at 11:34 am by FlatAssembler.)
(January 21, 2022 at 8:49 am)brewer Wrote: I don't share my birthday with anybody, the cake is all mine damnit.
This is a serious question, and I do not expect joke answers.
(January 21, 2022 at 10:04 am)arewethereyet Wrote: Another fascinating thread.
Are you being serious or ironic, I cannot tell?
