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Understanding the Birthday Problem: Probability and Applications in DNA Profiling

The Birthday Problem explores the probability that in a group of 'k' people, at least two share the same birthday, given 365 days in a year. The probability 'p(k)' quantifies this likelihood, derived as 'p(k) = 1 - p(k)', where 'p(k)' reflects the chance that all birthdays are distinct. Generalizations of this concept apply to finite sets, including hash functions in computer science where it assesses collision resistance. The practical application in DNA profiling, especially with short tandem repeats, reveals significant implications, such as a 50% probability of matching profiles among 120 individuals.

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Understanding the Birthday Problem: Probability and Applications in DNA Profiling

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  1. The Birthday problem and applications

  2. Calculating p(k) Assumptions : Every year contains 365 days People's birthdays are equally distributed over the 365 days of the year. p(k) is the probability that all k birthdays are different. p(2)= 364/365=1 −1/365 p(3)=(1 −1/365)*(1-2/365) k > 365 : p(k)=0 k<365 : p(k) is the probability that among k people, at least two will have the same birthday. p(k)=1-p(k) The Birthday problem

  3. p(k) : probability that among k people, at least two will have the same birthday p(k) : probability that all k birthdays are different

  4. Generalizations E : finite set p(k) : probability that among k elements of E, 2 elements are identical |E|= card(E)

  5. Hash function Definition : h = H(M) M is a variable-length message, h is a fixed-length hash value, H is a hash function. Requirement : Collisions-resistant: it should be hard to find two different messages m1and m2 such that H(m1)=H(m2) Let a hash function H have n-bit output (2 possible outputs). If we have k different messages, the probability that at least 2 messages are the same is Applications Input Output H key word n

  6. DNA The most prevalent method of DNA profiling used today uses short tandem repeats (STR) The system Quad STR analyse 4 genetic markers. The probability of a genetic profile is 0.01% The probability that among k people, at least two will have the same DNA profiling with this test is for k=120, the probability that at least 2 will have the same genetic profile is 50% . Vocabulary DNA profiling : Empreinte génétique

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