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Birthday Paradox

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How large a group would you need for better than 50/50 odds?

0-10
1
25%
10-20
1
25%
20-30
1
25%
30-40
1
25%
40-50
0
No votes
Greater than 50
0
No votes
 
Total votes: 4

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Reality Check
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Birthday Paradox

#1

Post by Reality Check »

Since there are two birthdays this week in my family it made me think of this math problem that I read about a long time ago. It goes like this:

How large a group of people do you need before there is a better than even chance that two people in the group will have the same birthday (month and day)? I am ignoring people born on February 29th for simplicity.

Can you solve it or get close without resorting to Google or the Internet?
fava
Posts: 4
Joined: Sat Mar 20, 2021 5:29 pm

Birthday Paradox

#2

Post by fava »

IIRC the formula is (364/365) * (363/365) * (362/365) * ... * ((365-n)/365)
and the answer becomes < 50% at n = 23

You can approximate it as 1.2 * sqrt(365)
W. Kevin Vicklund
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Joined: Tue Feb 23, 2021 4:26 pm

Birthday Paradox

#3

Post by W. Kevin Vicklund »

Close, the generic term is (366-n)/365 (because with n=1, there's a 100% chance that nobody shares a birthday since there is only one birthday)
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Reality Check
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Birthday Paradox

#4

Post by Reality Check »

From WIkipedia:
https://en.wikipedia.org/wiki/Birthday_problem

Yes, the correct answer to my question is 23. At a group size of 23 the probability that two people have the same birthday crosses 50% to 50.7%. For the average school class size of about 30 it is over 70%. It grows quickly as can be seen in this table and graph:
Screenshot 2023-12-21 at 09-34-14 Birthday problem - Wikipedia.jpg
Screenshot 2023-12-21 at 09-34-14 Birthday problem - Wikipedia.jpg (129.16 KiB) Viewed 2266 times
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