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The Birthday problem and applications

The Birthday problem and applications. 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.

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The Birthday problem and applications

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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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