Recitation 19 Fact from lecture. If there are N days in a year and m people in a room, then the proba bility that no two people in the room have the same birthday is about Problem 2. Suppose that we create aa national database of dNA profiles. Let's make some simplistic assumptions Each person can be classified into one of 20 billion different"DNA types".(For example, you might be type #13, 646, 572, 661 and the person next to you might be type #2, 785, 466,098 Let t(r)denote the type of person c Each dNa type is equally probable The dna types of Americans are mutually independent (a)a congressman argues that there are only about 250 million americans, so even if a profile for every American were stored in the database, the probability of even one coincidental match would be very small. How many profiles must the database actually contain in order for the probability of at least one coincidental match be bout 1/2 Solution. By the birthday principle, the probability of a match is around half when the number of entries is: √2ln2.20,000000006,511 (b) Person a is arrested for a crime that was committed by person y. At trial, jurors must determine whether x=y. The crime lab says a and y have the same dNA type. The prosecutor argues that the probability that r and y are different people is only 1 in 20 billion. Write the prosecutor's assertion in mathematical notation and explain her error Solution. The prosecutor is asserting that Pr(a and y are different people T(r)=r(y))=2.10- This assertion is at best false and arguably not even a well-formed mathematical statement. Either a and y are the same person or different people, regardless of the DNA types of all the people in the world. Thus, either c and y are the same person in every outcome or they are different people. Consequently, the probability above is either O or 1, but we don t know which The prosecutor's argument sounds confusingly similar to a correct assertion: if c and y are different people, then: Pr(T(x)=T(y))=2·1 The prosecutor can validly argue that either an amazing 1-in-20 billion coincidence involving DNA has occurred or else r and y are the same person On this basis, a jury might conclude that r is almost surely guilty, but there is nothing in our probabilit model to justify that conclusion directl� Recitation 19 3 Fact from lecture. If there are N days in a year and m people in a room, then the proba￾bility that no two people in the room have the same birthday is about: e−m2/(2N) Problem 2. Suppose that we create a a national database of DNA profiles. Let’s make some simplistic assumptions: • Each person can be classified into one of 20 billion different “DNA types”. (For example, you might be type #13,646,572,661 and the person next to you might be type #2,785,466,098.) Let T(x) denote the type of person x. • Each DNA type is equally probable. • The DNA types of Americans are mutually independent. (a) A congressman argues that there are only about 250 million Americans, so even if a profile for every American were stored in the database, the probability of even one coincidental match would be very small. How many profiles must the database actually contain in order for the probability of at least one coincidental match be about 1/2? Solution. By the birthday principle, the probability of a match is around half when the number of entries is: 2 ln 2 20 · , 000, 000, 000 = 166, 511 (b) Person x is arrested for a crime that was committed by person y. At trial, jurors must determine whether x = y. The crime lab says x and y have the same DNA type. The prosecutor argues that the probability that x and y are different people is only 1 in 20 billion. Write the prosecutor’s assertion in mathematical notation and explain her error. Solution. The prosecutor is asserting that: 10−10 Pr (x and y are different people | T(x) = T(y)) = 2 · This assertion is at best false and arguably not even a well­formed mathematical statement. Either x and y are the same person or different people, regardless of the DNA types of all the people in the world. Thus, either x and y are the same person in every outcome or they are different people. Consequently, the probability above is either 0 or 1, but we don’t know which. The prosecutor’s argument sounds confusingly similar to a correct assertion: if x and y are different people, then: Pr (T(x) = T(y)) = 2 10−10 · The prosecutor can validly argue that either an amazing 1­in­20 billion coincidence involving DNA has occurred or else x and y are the same person. On this basis, a jury might conclude that x is almost surely guilty, but there is nothing in our probability model to justify that conclusion directly