#birthday problem

[verysharpfish]

Mfer let another guy publish the finding cause he was like "Literally no way someone else didn't already work this out"

[math1089]

40 Interesting Facts About Maths

40 Interesting Facts About Maths

We could use up two Eternities in learning all that is to be learned about our own world and the thousands of nations that have arisen and flourished and vanished from it. Mathematics alone would occupy me eight million years. Mark Twain Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy…

View On WordPress

[ectoplasmic-hearts]

So my birthday is ending with me crying cause my own father though it more important to go out drinking than spend my birthday with me!! 😃😅😔😢

[lawinfosysfci-blog]

More on the Population-wide Database

Chris Halikes, posted a comment on yesterday's blog posting about expanding the DNA database to the entire population. He expressed the following concern:

The mathematics of coincidental matches, particularly with respect to the size of the database, is a thorny subject. However, the odds of a coincidental match seem to rise (the so-called birthday problem). Given the strength of the tunnel vision that sets in when DNA is involved (the Lukis Anderson/Raveesh Kumra case is a recent example), I would think long and hard about the wisdom of having an all-inclusive database.

The prospect of more false accusations definitely deserves thought. In evaluating this risk, the birthday problem is not on point. In the birthday problem, the number of comparisons grows exponentially with the number of people of in the room because there is no single birthday of interest. For a group of size N, there are approximately N2/2 comparisons. For a fixed birthday, however, there are only N comparisons.  In case work, no one examines all possible pairs of profiles. Like a fixed birthday, a single crime-scene profile is compared to the profiles in the database. Because the number of comparisons is N, the risk grows only linearly with the size of the database. The exponential growth in the Birthday Paradox does not occur. CODIS is expanding the number of loci in the profile, so the chance that two people (other than monozygotic twins) share the same profile will be even smaller than it is now. When the crime-scene sample yields a good profile (say, 16 or more loci), I cannot see why increasing the database size to that of the entire population would produce many matches to people whose DNA was not at the crime-scene. The record of an innocent monozygotic twin would pop up in every database trawl for a crime-scene DNA profile that actually belongs to the guilty twin. Strictly speaking, these are not these "coincidental," because there is a deterministic explanation. For such matches -- and for truly coincidental ones -- the population-wide database flags the problem for the police. They get matches to more than one individual! They will know that they must investigate further. Thus, universality alleviates the problem of a "coincidental match" by making every such match apparent. The problem in the Kumra case, according to prosecutors, was secondary transfer to fingernails that (I assume) did not have DNA from the actual killers. Likewise, if the police or anyone else plants DNA from a target in an incriminating place where the perpetrator's DNA might be found -- and the perpetrator's DNA is not there -- the database trawl will identify the target instead of the perpetrator. If a crime-scene sample becomes contaminated with extraneous DNA from individuals who were not present at the crime-scene  in an amount sufficient to yield a clear and complete profile, the result could be a false accusation and ensuing conviction for people living in the vicinity, being in the correct age range and physical capacity, and lacking a convincing alibi. Another limitation of DNA databases is that, at best, they can only show that an individual was at a crime-scene at some point. If police and prosecutors unreflectively equate presence with guilt and a suspect has no persuasive explanation for his presence, injustice could follow. However, this is a problem today. Arguably, a universal database might diminish the problem: cases of innocent presence would arise more often, leading to greater sensitivity to this limitation. Less worrisome are errors such as mislabeling or mistyping the samples in the database. If the profile recorded for me is not my profile but was present at the crime-scene, this mistake will become apparent when I am retested after the cold hit, as is standard procedure. The confirmatory test will exclude me. A population-wide has other advantages than those I mentioned today and yesterday -- but it also has its share of disadvantages. References

Persi Diaconis & Frederick Mosteller, Methods for Studying Coincidences, 84 J. Am. Stat. Ass’n 853 (1989) (discussed in Gina Kolata, 1-in-a-Trillion Coincidence, You Say? Not Really, Experts Find, N.Y. Times, Feb. 27, 1990, http://www.nytimes.com/1990/02/27/science/1-in-a-trillion-coincidence-you-say-not-really-experts-find.html?pagewanted=all&src=pm)

David H. Kaye, Beyond Uniqueness: The Birthday Paradox, Source Attribution, and Individualization in Forensic Science Testimony, 12  Law, Probability & Risk 3 (2013)

Related Postings

David H. Kaye, On the Hypothetical Population-wide Database, Forensic Science, Statistics & the Law, July 29, 2013, http://for-sci-law-now.blogspot.com/2013/07/on-hypothetical-population-wide-dna.html

-----, Good Point, Bad Math: DNA Database Statistics Misunderstood (Again), July 26, 2013, http://for-sci-law-now.blogspot.com/2013/07/good-point-bad-math-dna-database.html

[lawinfosysfci-blog]

Good Point, Bad Math: DNA Database Statistics Misunderstood (Again)

In a New York Times op-ed article, High-Tech, High-Risk Forensics, Hastings Law School Professor Osagie K. Obasogie mentions a false arrest in the investigation of the murder near San Jose of millionaire investor Raveesh Kumra. A database trawl seemed to implicate Lukis Anderson, a homeless man who then “spent more than five months in jail with a possible death sentence hanging over his head” before prosecutors received records showing that Mr. Anderson was in a hospital for alcohol intoxication on the night of the murder. Prosecutors have suggested that paramedics who treated Anderson for intoxication at a liquor store in San Jose a few hours before the murder inadvertently transferred his DNA to Mr. Kumra’s fingernails. This is by no means the first instance of secondary transfer of DNA to a crime scene (if that is the correct explanation), and forensic scientists have been conducting studies to see how often such transfer occurs. The case shows that police and prosecutors, as well as defense counsel and jurors, should not be overawed by matches from DNA database trawls. This is the good point that Professor Obasogie makes. The rest of the op-ed article perpetuates mathematical fallacies about DNA databases. First, Professor Obosagie questions “the frequent claim that it is highly unlikely, if not impossible, for two DNA profiles to match by coincidence.” Nothing is impossible, but not many people share the same DNA identification profiles. The op-ed tries to deny this with the observation that “[a] 2005 audit of Arizona’s DNA database showed that, out of some 65,000 profiles, nearly 150 pairs matched at a level typically considered high enough to identify and prosecute suspects. Yet these profiles were clearly from different people.” The 150 or so matches were, in fact, mismatches. That is, they were partial matches that actually excluded every “matching” pair. Only if an analyst improperly ignored the nonmatching parts of the profiles or if these did not appear in a crime-scene sample could they be reported to match. Moreover, even we treat all 150 partial matches as tantamount to false full matches in casework, a proper analysis must account for artificially pairing every profile with every other profile and for the many ways to find some kind of partial match. The scientific and legal literature clearly shows that many partial matches are to be expected under these conditions. The 65,000-some samples then in the database gave rise to over a trillion possible partial matches. (The same combinatorial explosion explains the well-known “Birthday paradox” in probability theory.) A mere 150 partial matches out of a trillion opportunities to make such matches represents a quasi-false match rate on the order of about 100 per trillion (0.0000000001). This number is not quite zero, but neither is it “high risk.” Second, Professor Obosagie notes that “There are also problems with the way DNA evidence is interpreted and presented to juries.” True enough. A common problem is the confusion of a random match probability with a source probability, and the op-ed makes this very mistake when it claims that jurors in the San Francisco prosecution of John Puckett five years ago were “told that there was only a one-in-1.1 million chance that this DNA match was pure coincidence.” According to Mr. Puckett’s brief on appeal, the DNA analyst testified not to this source probability, but only that “the profile would occur at random among unrelated individuals in about 1 in 1.1 million U.S. Caucasians ... .” (The op-ed also mistakenly asserts that the defendant, who died before his appeal was heard, “is now serving a life sentence.”) How prosecutors should present statistics in the cases in which the match came about from a database trawl is an important question, but it would have been misleading to tell jurors that the “chance that this DNA match [to John Puckett] was pure coincidence” was “one in three,” as Puckett wanted to do. Putting the possibility of laboratory error to the side (as the op-ed does), the probability of a database trawl suggesting that John Puckett was the killer of 22-year-old Diana Sylvester is 1 if Puckett was in fact the killer. It is about 1 in 1.1 million if he was not. A figure like 1 in 3 therefore is not suitable for presenting to a jury as a measure of how revealing a database match is, but perhaps it is useful for another purpose. Puckett produced the probability by multiplying the tiny random-match probability of 1 in 1.1 million by the size of the database. This multiplication yields an estimate of how often trawling a database populated entirely by people innocent of every crime for which the database is used would produce matches to anyone. The larger the number, the greater the risk that “innocent databases”—those that fail to contain profiles of the individuals leaving their DNA at crime-scenes—will lead to false accusations. Of course, we know that this is not a meaningful estimate of the false positive rate of the real databases, for most matches are corroborated with other evidence. If all databases were innocent, and the 1-in-3 number applied, we would be seeing lots of matches to people too young or too old to have committed the crime being investigated, in prison at the time, or having other solid alibis like Mr. Anderson’s. Thus, it is improbable that strictly coincidental database matches are common—particularly in the run-of-the-mill cases in which the crime-scene DNA is not a mixture or too small or too degraded to be fully typed. Nevertheless, no one can say exactly how often DNA database searches turn up plausible suspects who are, in fact, innocent. This takes me back to Professor Obasogie’s one good point—let’s not get carried away with DNA database matches. There can be innocent explanations for reported matches. But let’s not become overly skeptical of the value of DNA databases to generate investigative leads—and let’s not use bad math to prop up our conjectures. References

Osagie K. Obasogie, High-Tech, High-Risk Forensics, N.Y. Times, July 25, 2013, at A27

D. J. Daly et al., The Transfer of Touch DNA from Hands to Glass, Fabric and Wood, 6 Forensic Sci. Int’l Genetics 41 (2012)

David H. Kaye, Trawling DNA Databases for Partial Matches: What Is the FBI Afraid Of?, Cornell Journal of Law and Public Policy, Vol. 19, No. 1, Fall 2009, pp. 145-171

----- , Rounding Up the Usual Suspects: A Legal and Logical Analysis of DNA Database Trawls, North Carolina Law Review, Vol. 87, No. 2, January 2009, pp. 425-503

-----, Rehash and Mishmash in the Washington Monthly, Double Helix Law, Feb. 27, 201

Carll Ladd et al., A Systematic Analysis of Secondary DNA Transfer, 44 J. Forensic Sci. 1270 (1999)

Alex Lowe et al., The Propensity of Individuals to Deposit DNA and Secondary Transfer of Low Level DNA from Individuals to Inert Surfaces, 129 Forensic Sci. Int’l 25 (2002)

S. Malsom et al., The Prevalence of Mixed DNA Profiles in Fingernail Samples Taken from Couples Who Co-habit Using Autosomal and Y-STRs, 3 Forensic Sci. Int’l Genetics 57 (2009)

[thehalfbloodpsych]

Hey guys!! My 25th birthday came up super quick this year and I don't have anything planned because it's on a Tuesday! But I'm thinking of celebrating a bit on the 14th (Sunday before my birthday) and I'm not sure how!! Anyone have any cheap-ish birthday ideas for me? 😬😁 anything would be appreciated!! Thanks

[sontzuuu]

Umm. So i asked that mom and stepdad that they would not bring cake for me. And now they did and it’s princess cake that i don’t personally like (because i can’t stand marzipan). And now mom got some kind of whining scene that i don’t like this cake that i used to like years and years ago. Like what. Calm down. I did not ask for this cake so ..... why is it here? And how is it my fault? i did not even want this freaking cake! 

[intergalacticgazette]

The Birthday Problem and Expected Shared Birthdays

So most math majors/mathematicians (I think) have heard the birthday problem, also known as the birthday paradox. If you haven’t, it is a probability problem that demonstrates an unexpected answer. The simple summary is that given 23 random people in a room, there is a greater than 50% chance that two people in the room share a birthday(month and date).

Recently, I have been saying happy birthday to many friends on Facebook. The number of people is usually one, but recently it has been two or three people sharing a birthday.

This made me wonder how many Facebook friends I had shared a birthday with me or someone else on my friends list. I took a count, scrolling through the birthdays listed on Facebook. I will tell you the number later.

The birthday problem and my Facebook friends made me wonder what the expected number of shared birthdays is among a group of people. I decided to approach finding the solution with simulation using Mathematica.

I started by creating a system to generate a random date for a set number of friends. The RandomInteger function worked well for this purpose. It selects a number from 1 to 365 to cover all the days in a typical year a person can be born. The number is selected and added to a list Dates.

My next step was to count how many duplicates were in the Dates list. My initial approach was a mess of For and If loops. It would state that there was one shared date if the same number appeared twice in Dates, but 3 shared dates if the same number appeared 3 times in Dates. Since it takes two or more people to share a date, I needed a number appearing twice to count twice to count twice, appearing 3 times to count 3 times, and so on. Finding the Count function shortened and simplified my approach dramatically.

Having the random date generation and shared date counting working together, I needed to repeat the process. A Do loop worked well for this. My warning on this is how many times you run the Do loop or displaying SharedDays. SharedDays is the list of how many shared birthdays appeared in a generated list of Dates. 

Using a reasonable number of sample runs, I used the Tally function on the SharedDays list to generate a histogram and took the mean of SharedDays. 

This simulation showed that given 153 people with random birthdays, the average number of people sharing birthdays is 52 people. The decision to simulate 153 people is from the fact that I have 152 Facebook friends and myself. My 152 Facebook friends and I have 49 people sharing birthdays. The standard deviation(not shown) is about 7. So my Facebook friends and I are within one deviation for number of people sharing a birthday.