Birthday problem: Difference between revisions
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Using simulation, estimate the number of independent people required in a groups before we can expect a ''better then even chance'' that at least 2 independent people in a group share a common birthday. Furthermore: Simulate and thus estimate when we can expect a ''better then even chance'' that at least 3, 4 & 5 independent people of the group share a common birthday. For simplicity assume that all of the people are alive... |
Using simulation, estimate the number of independent people required in a groups before we can expect a ''better then even chance'' that at least 2 independent people in a group share a common birthday. Furthermore: Simulate and thus estimate when we can expect a ''better then even chance'' that at least 3, 4 & 5 independent people of the group share a common birthday. For simplicity assume that all of the people are alive... |
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'''Suggestions for improvement''': |
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* Converging to the nᵗʰ solution using a root finding method, as opposed to an using an extensive search. |
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* Kudos (κῦδος) for finding the solution by proof (in a programming language) rather then by construction and simulation. |
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'''See also:''' |
'''See also:''' |
Revision as of 07:14, 4 June 2013
This page uses content from Wikipedia. The current wikipedia article is at Birthday Problem. The original RosettaCode article was extracted from the wikipedia article № 296054030 of 21:44, 12 June 2009 . The list of authors can be seen in the page history. As with Rosetta Code, the pre 5 June 2009 text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
In probability theory, the birthday problem, or birthday paradox This is not a paradox in the sense of leading to a logical contradiction, but is called a paradox because the mathematical truth contradicts naïve intuition: most people estimate that the chance is much lower than 50%. pertains to the probability that in a set of randomly chosen people some pair of them will have the same birthday. In a group of at least 23 randomly chosen people, there is more than 50% probability that some pair of them will both have been born on the same day. For 57 or more people, the probability is more than 99%, and it reaches 100% when the number of people reaches 366 (by the pigeon hole principle, ignoring leap years). The mathematics behind this problem leads to a well-known cryptographic attack called the birthday attack.
Task:
Using simulation, estimate the number of independent people required in a groups before we can expect a better then even chance that at least 2 independent people in a group share a common birthday. Furthermore: Simulate and thus estimate when we can expect a better then even chance that at least 3, 4 & 5 independent people of the group share a common birthday. For simplicity assume that all of the people are alive...
Suggestions for improvement:
- Estimating the error in the estimate to help ensure the estimate is accurate to 4 decimal places.
- Converging to the nᵗʰ solution using a root finding method, as opposed to an using an extensive search.
- Kudos (κῦδος) for finding the solution by proof (in a programming language) rather then by construction and simulation.
See also:
ALGOL 68
File: Birthday_problem.a68<lang algol68>#!/usr/bin/a68g --script #
- -*- coding: utf-8 -*- #
REAL desired probability := 0.5; # 50% #
REAL upb year = 365 + 1/4 # - 3/400 but alive, ignore >113yo #, INT upb sample size = 10000,
upb common = 5;
FORMAT name int fmt = $g": "g(-0)"; "$,
name real fmt = $g": "g(-0,4)"; "$, name percent fmt = $g": "g(-0,2)"%; "$;
printf((
name real fmt, "upb year",upb year, name int fmt, "upb common",upb common, "upb sample size",upb sample size, $l$
));
INT required common := 1; # initial value # FOR group size FROM required common WHILE required common <= upb common DO
INT sample with no required common := 0; TO upb sample size DO # generate sample # [group size]INT sample; FOR i TO UPB sample DO sample[i] := ENTIER(random * upb year) + 1 OD; FOR birthday i TO UPB sample DO INT birthday = sample[birthday i]; INT number in common := 1; # special case = 1 # IF number in common >= required common THEN found required common FI; FOR birthday j FROM birthday i + 1 TO UPB sample DO IF birthday = sample[birthday j] THEN number in common +:= 1; IF number in common >= required common THEN found required common FI FI OD OD # days in year #; sample with no required common +:= 1; found required common: SKIP OD # sample size #; REAL portion of years with required common birthdays = (upb sample size - sample with no required common) / upb sample size; print("."); IF portion of years with required common birthdays > desired probability THEN printf(( $l$, name int fmt, "required common",required common, "group size",group size, # "sample with no required common",sample with no required common, # name percent fmt, "%age of years with required common birthdays",portion of years with required common birthdays*100, $l$ )); required common +:= 1 FI
OD # group size #</lang>Output:
upb year: 365.2500; upb common: 5; upb sample size: 10000; . required common: 1; group size: 1; %age of years with required common birthdays: 100.00%; ...................... required common: 2; group size: 23; %age of years with required common birthdays: 50.09%; ................................................................ required repeat: 3; group size: 87; %age of years with required repeat birthdays: 50.37%; ................................................................................................... required repeat: 4; group size: 186; %age of years with required repeat birthdays: 50.70%; ............................................................................................................................... required repeat: 5; group size: 313; %age of years with required repeat birthdays: 50.92%;