birthday paradox – a simulation of shared birthday experiments

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Overall Comment: "While the paper is well written, it tried to fit too much trivial detail in at the expense of material which was demoted to the appendices. The paper should be able to stand on its own without the appendices; the appendices simply add to/support the paper. code and extracted Javadoc very impressive."Abstract: Being one of the most well-known and famous problems inprobability1, the question is asked: How many randomly chosen peopleare needed to achieve at least a 50% probability that some pair will bothhave been born on the same day? Since the chance of any two personshaving the same birthday is remote, many of us would expect this numberto be rather large. However, it turns out that this is not the case, andhence the paradox.A simulation of empirical testing and results was conducted to simulatemultiple trials, where people of a given group size have their birthdayscompared. The probability is consistently monitored and established asthe number of successful experiments as a proportion of the total numberof experiments performed.As the theoretical value is already known (23), it was also the goal-drivingresult for the algorithm being implemented in Java, and happily, wasachieved.

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BIRTHDAY PARADOX A SIMULATION OF SHARED BIRTHDAY EXPERIMENTSStephen Hogan (50217631) Postgraduate Diploma in IT (Evening) Dublin City University School of Computinghogans2@mail.dcu.ie

Abstract Being one of the most well-known and famous problems in probability1, the question is asked: How many randomly chosen people are needed to achieve at least a 50% probability that some pair will both have been born on the same day? Since the chance of any two persons having the same birthday is remote, many of us would expect this number to be rather large. However, it turns out that this is not the case, and hence the paradox. A simulation of empirical testing and results was conducted to simulate multiple trials, where people of a given group size have their birthdays compared. The probability is consistently monitored and established as the number of successful experiments as a proportion of the total number of experiments performed. As the theoretical value is already known (23), it was also the goal-driving result for the algorithm being implemented in Java, and happily, was achieved.

2. BACKGROUNDMathematical ModelOne of the basic rules of probability: the sum of the probability that an event will happen and the probability that the even will not happen is always 1. In other words, the chance that anything might or might not happen is always 100%. If we can work out the probability that no two people will have the same birthday, we can use this rule to find the probability that two people will share a birthday: P(event happens) + P(event does not happen) = 1P(two people share birthdays) + P(no two people share birthdays) = 1 P(two people share birthdays) = 1 P(no two people share birthdays) Comment [JM5]: Mixed font size. Comment [JM1]: Keep it formal and to the point. Comment [JM4]: This could be compressed by simply giving a single equation and pointing the reader to a reference whre more detail is given.

1. INTRODUCTIONPremise born in 19382: In a room of just 23 people, there is a 50% probability of two of these people sharing the same birthday, (ignoring years of birth). In a room of 75 people, there is a 99.9% chance of two people with matching birthdays. This is one particular case of exponential (Saliusian) sets, where duplicates are allowed. Exponents are not intuitive, and thus why our linear-thinking leads us to an incorrect estimation! While theoretical mathematical models and proofs have been derived in previous works outside the scope of this paper, the object of this paper is not to put the formulae into action; moreover the objective of this paper is to highlight a proof by example; i.e. that a simulation of birthdates for a group of people is analysed for this probability result, based on the aforementioned premise. For the following two sections, Background and Method, to be presented here, both share the following assumptions: That there are only 365 days in a year, i.e. thus ignoring leap years. This also results in ignoring the suspension of leap day on years divisible by 100 that are also divisible by 400. Birth years are ignored. Peoples birthdays are equally distributed throughout the year; (i.e. influencing elements such as seasonality are not factored in). Obviously in real-life, birthday distributions are not uniform, i.e. not all dates are equally likely. The date of a persons birthday does not affect the date of another person birthday, i.e. twins, triplets, etc.

The formula for the probability that n people have different birthdays (month and day) is3: 365 (1) 365 n! * 365n Therefore, the probability that at least two of them share the same birthday is: 365 1 (2) 365 n! * 365 n Having (2) graphed in Figure 1 it is clearly seen where, at the probability of 50%, cross-referencing it with the number of people reveals a value of 23:

Comment [JM2]: Use a reference rather than a footnote.

Comment [JM3]: A reference would be welcome here.

Figure 1: A graph showing the approximate probability of at least two people sharing a birthday amongst a certain number of people 4.

This is not a paradox in the literal sense it just highlights the fact that people expect the value to be much larger. 2 American Mathematical Monthly in 1938 in Zoe Emily Schnabel's The estimation of the total fish population of a lake, under the name of capture-recapture statistics.1

As Dr Math FAQ The Birthday Problem (http://mathforum.org/dr.math/faq/faq.birthdayprob.html) 4 Wikipedia - Birthday Problem (http://en.wikipedia.org/wiki/Birthday_paradox)3

3. METHODProgramming MethodologyThe simulation was written in the Java language 5. A text file that lists multiple trials with the following parameters serves as input to the program, (defaults outlined here are geared to solving the problem in question): Number of matches to be checked (default: K = 2) Number of trials (various >> values) Starting group size (default: N = 2) Group size increment (default: 1) Terminating probability (default: 0.5) Assumptions: The group size will never be greater than 1,000. In the case that the starting value of N is less than K, you should start the simulation with N = K. The reason is simple: How do you possibly find three (i.e. K) matches in a group of two (i.e. N) persons? Beginning with group size of N = 2 people, we initialise an array with the random birthdays of N = 2 people; (the random number generator is being seeded with the current time). We compare every pair wise (number of matches K = 2) combination of people in the group of N = 2 and check the existence of any two persons having the same birthday. This will be repeated number of trials times with different groups of two people. If the average occurrence of two persons having the same birthday in these one thousand trials exceeds 0.5 (i.e. the probability P), the simulation terminates. Otherwise, N is incremented by group size increment = 1, and the entire simulation of one thousand trials is repeated with randomized groups of (starting group size = 2) + (group size increment = 1) people. The flexibility in reading in values from a file that control the execution of the algorithm allows us to evaluate numbers of people for various probabilities and/or enumerate possible combinations of more than two persons, differing numbers of trials and matches, and check for shared birthdays, for example. A review of the Java source code in Appendix 2 should reveal other variations. For each trial, a number of random birthdays are generated and placed into an array; (here, we use the Julian Date format, 1365). These birthdays are sorted and then iterated through to find the same values in consecutive elements in the array, denoting a success. Once the trials have completed running, the probability is evaluated as6: currProbability = numSameBirthday / numTrials

The last record in this table is graphed in Figure 2. Preparing graphs for all records processed in this table (Appendix 4) reveals, interestingly, that time performance dips in proportion to the curve of probability when tending to 23 people, irrespective of the number of trials:Simulation 80.60000 25000

0.50000 20000

0.40000Probability

15000 0.30000 10000 0.20000Time (ms)

5000 0.10000

0.00000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 No. People Probability Time (ms)

0

Figure 2: Simulation 8

Comment [JM9]: You need to explain what Simulation 8 is.

5. CONCLUSIONS & RECOMMENDATIONSThis research reports on a simulated empirical study of the Birthday Paradox. The findings suggest that there is a strong similarity between the theoretical and (simulated) actual probabilities. Specifically, that for any group size or number of trials, to achieve at least a 50% probability in solving the problem, we would need at least 23 people in a room for comparison (with previously unknown birthdays). The input file allows for variations on the number of people > pairs as well as the group increment size to be > 1, as well as other terminating probabilities. A further recommendation would be to refine the sorting algorithm even further, or adopt a faster mechanism of sorting, especially for large N and K. Parallelisation of Quicksort, for example, would be ideal, as synchronisation is not a requirement. Java Threading would be an approach for this. Another approach to solving this problem would be to base it on collisions by tracking as each person enters the room and checking to see if there is a match with any other person. An array of 365 elements would only be needed, and a random date only generated for each person until either the entire group size is exhausted or a match has been found. The author of this paper decided against this approach after initially selecting it, as it would not be possible to measure time performance as fluidly.

Comment [JM10]: We Comment [JM6]: Why not use algrbraic notation as in (1) and (2). Also: This eqn is not numbered.

4. RESULTS & DISCUSSIONWith the availability of having an input file, multiple case scenarios can be generated. Table 1 is an example of sample data was available on the input file, (as per aforementioned format in Section 3):2 2 2 2 2 2 2 2 10000 20000 30000 50000 700000 1000000 2000000 5000000 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

6. REFERENCES[1] Birthday Paradox Wikipedia.com (http://en.wikipedia.org/wiki/Birthday_paradox) [2] CS1101C Lab 3 Birthday Paradox National University of Singapore, School of Computing (http://www.comp.nus.edu.sg/~cs1101cl/labs_sem2_0405/lab3/oddweek/) [3] How to Generate Random numbers About.com (http://java.about.com/od/javautil/a/randomnumbers.htm) [4] Quick Sort Implementation with median-of-three partitioning and cutoff for small arrays Java-Tips.org (http://www.java-tips.org/java-se-tips/java.lang/quic