4 It’s My Birthday Too, Steven Strogatz - Six part essay on math

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//-->It’s My Birthday Too, Yeah — opinionator.blogs.nytimes.com — Readability2013-08-13opinionator.blogs.nytimes.comRead LaterIt’s My Birthday Too, Yeahby STEVEN STROGATZ • Oct. 1, 2012 •originalBy an amazing coincidence my sister, Cathy, and my Aunt Vere have the same birthday:April 4.Actually, it’s not so amazing. In any extended family with enough siblings, aunts, unclesand cousins, you’d expect at least one such birthday coincidence. Certainly, if there are366 people in the family — more relatives than days of the year — they can’tallhavedifferent birthdays, so a match is guaranteed in a family this big. (Or if you’re worriedabout leap year, make it 367.)But suppose we don’t insist on absolute certainty. A classic puzzle called the “birthdayproblem” asks: How many people would be enough to make the odds of a match at least50-50?The answer, just 23 people, comes as a shock to most of us the first time we hear it. Partlythat’s because it’s so much less than 366. But it’s also because we tend to mistake thequestion for one aboutourselves. Mybirthday.John Allen Paulos gave a vivid example of this error in his trenchant best seller“Innumeracy”:A couple of years ago, someone on the Johnny Carson show was trying to explain [why 23is the answer to the birthday problem]. Johnny Carson didn’t believe it, noted that therewere about 120 people in the studio audience, and asked how many of them shared hisbirthday of, say, March 19. No one did, and the guest, who wasn’t a mathematician, saidsomething incomprehensible in his defense.For years I’ve been dying to see a clip of that scene. It’s become legendary, an iconicmoment in mathematical pop culture. Yet I couldn’t find it on YouTube, and none of mycolleagues, including Paulos, could remember when it occurred or who the humiliatedguest was. Memories have understandably faded over the years.But fortunately, as of 2010 every surviving tape of “The Tonight Show With JohnnyCarson” has been digitized and made publicly available. Thanks to the Carson archivesand some ace detective work by a researcher at The Times, we can now watch what reallyhttp://www.readability.com/articles/l2vscn6p1/6It’s My Birthday Too, Yeah — opinionator.blogs.nytimes.com — Readability2013-08-13happened when Johnny met the birthday problem. The encounter offers lessons not justabout math but about memory as well.Before we get to the clip, you may be wondering why the birthday problem is worthstudying. For one thing, it highlights how wrong our intuition can be about coincidences,and how easily we underestimate the power of chance — a cautionary lesson for anyoneworking in sports, finance or any other field where fluky things happen a lot.Second, the reasoning used to solve the birthday problem is transferable. Medicalstatisticians use it to estimate the likelihood of finding matches within pools of potentialtransplant donors and recipients on such characteristics as blood type, Rh factors andother immunological markers. Criminologists use it to calculate how many partialmatches between DNA profiles they should expect to find — by chance alone — whentrawling databases of convicted offenders. Cryptographers use it to analyze malicious“birthday attacks” that adversaries can deploy to subvert digital signatures.To work our way toward solving the birthday problem, let’s first simplify it by assumingthere are 365 days in the year, and that all birthdays are independent and equally likely.Under these assumptions, we want to figure out how many people we need for the odds tobe better than 50-50 that at least two of them have the same birthday.The solution relies on a single principle, used over and over. I’ll call it the combinationprinciple, but it’s just common sense. Anyone who gets dressed in the morning knows it.Suppose you have 3 pairs of pants and 5 shirts. (I realize you probably have more thanthis, but pretend you’re a math professor.) How many different outfits can you create?(And don’t worry if some of the shirts and pants don’t go too well together — remember,you’re a math professor!) Say you decide to wear your ratty blue jeans. Then with fiveshirts to choose from, that gives you 5 outfits right there. Or you could wear those nicepolyester khakis you still have from your high school graduation. Combine them with anyof the five shirts and that’s another 5 outfits. Finally, you could go casual and wear yourStar Trek sweat pants along with any of the five shirts, creating 5 more outfits andbringing the total to 3 times 5, or 15, outfits in all.That’s the combination principle in action: If you can makeMchoices of one thing (like 3pairs of pants) andNchoices of another (like 5 shirts), you can makeMxNcombinationsof them both (15 outfits). The principle also extends to more than two things. If you wantto top off your outfit with a stylish hat and you have 6 to choose from, you can create 3 x 5x 6 = 90 ensembles of pants, shirts and hats.Next, let’s apply this principle to a warm-up birthday problem featuring the first threeUnited States presidents. Relative to the New Style (Gregorian) calendar, Georgehttp://www.readability.com/articles/l2vscn6p2/6It’s My Birthday Too, Yeah — opinionator.blogs.nytimes.com — Readability2013-08-13Washington was born on Feb. 22, John Adams on Oct. 30, and Thomas Jefferson on April13. Unsurprisingly, no matches. To figure out the odds of this happening by chance, weimagine alternate realities — all the possible combinations of birthdays thatcouldhaveoccurred — and then calculate what fraction of those combinations involve three distinctbirthdays.According to the combination principle, there are 365 x 365 x 365 combinations of threebirthdays, since any day of the year is possible for each of the three presidents. To counthow many of these combinations contain no matches, let Washington go first. He has all365 days at his disposal. But once his birthday is fixed, he leaves Adams with only 364choices to avoid a match, which in turn leaves Jefferson only 363. So, by the combinationprinciple, there are 365 x 364 x 363 nonmatching combinations of three birthdays, out ofa total of 365 x 365 x 365 combinations altogether.Hence the probability that all three birthdays differ is the ratio of these huge numbers:which is about 0.9918 or 99.18 percent. In other words, it was almost a sure thing thatnone of the birthdays would match, as we’d expect by common sense.To extend this result to four or more people, look again at the fraction above and savor thepatterns in it. For three people the fraction has three descending numbers — 365, 364, 363— in the numerator, and three copies of 365 in the denominator. So for four people, thenatural and correct guess is that the answer becomesThis expression is merely the fraction we found earlier for three people, multiplied by362/365. Doing the arithmetic then gives 0.9836, or a 98.36 percent chance that fourrandom people have four different birthdays. That means the probability that two or moreof themsharea birthday is about 1 – 0.9836 = 0.0164, or 1.64 percent.Continuing in this way, ideally with the help of a spreadsheet, computer oronline birthdayproblem calculator,we can crank out the corresponding probabilities for any number ofpeople. The calculations show that the odds of a match rise sharply as the group getslarger. With 10 people, the odds are almost 12 percent; with 20 people, 41 percent. Whenwe reach the magic number of 23 people, the odds climb above 50 percent for the firsttime, which is what we were trying to prove.Intuitively, how can 23 people be enough? It’s because of all the combinations they create,all the opportunities for luck to strike. With 23 people, there are 253 possiblepairsofhttp://www.readability.com/articles/l2vscn6p3/6It’s My Birthday Too, Yeah — opinionator.blogs.nytimes.com — Readability2013-08-13people (see the notes for why), and that turns out to be enough to push the odds of amatch above 50 percent.Incidentally, if you go up to 43 people — the number of individuals who have served asUnited States president so far — the odds of a match increase to 92 percent. And indeedtwo of the presidentsdohave the same birthday: James Polk and Warren Harding wereboth born on Nov. 2.And now… here’s Johnny! On Feb. 6, 1980, about 14 minutes into “The Tonight Show,”Johnny Carson and his sidekick Ed McMahon begin bantering about Ed’s upcomingbirthday and famous Americans born in February. Then Johnny changes gears and says,“You know, I’m gonna try something tonight. Now I may get this wrong. I remember thisfrom a long time ago.” He tries to pose the birthday problem, fumbles a bit with thephrasing, and finally comes up with this formulation: “How many people would you thinkwould have to be in a room that your odds would be almost sure that they would haveexactly the same birthday on the same day?”Ed guesses 1,000, which suggests he doesn’t understand the question, since 366 peoplealready ensure a match. Johnny shakes his head slightly and says, “Something like 35 or40.” “That’s all?” Ed interjects, surprised, and Johnny continues, “the odds are prettygood, if I remember.”You’ve got to appreciate what a bombshell this is, at least in my world: Contrary to thecollective memory of the mathematical community, itwasn’ta guest scientist who posedthe puzzle that night — it was Johnny himself! Though in retrospect, our mistake waspredictable. As with many other false memories and urban legends, our garbledrecollection makes more sense than what actually happened. We don’t normally think ofJohnny as someone who’d pose a classic math problemandknow its surprising answer.Then comes the next twist: Ed leads Johnny astray. He says, “Pick a day and see if we haveit.” Johnny says, “Lady in the front row, what’s your birthday?” “August 9,” she says.“Anyone else here have a birthday on August 9?” asks Johnny. “No?” He looks baffled. “Wehave 500 people here.” They try someone else’s birthday, April 9, and again there are nomatches. Ed acts vindicated and the audience finds the whole thing hilarious.What went wrong? Johnny did the wrong experiment! By asking for matches specificallyto August 9 (or to April 9), he altered the problem. The original problem asked for amatch betweenanytwo people onanyday. A much larger number of people (253, as itturns out) are required for there to be a better than even chance of finding a match to aspecificbirthday like Aug. 9.Let me leave you with one last coincidence to ponder. John Adams and Thomas Jeffersondied on the same day — the Fourth of July, 1826, exactly 50 years after the adoption of thehttp://www.readability.com/articles/l2vscn6p4/6It’s My Birthday Too, Yeah — opinionator.blogs.nytimes.com — Readability2013-08-13Declaration of Independence.Math can’t explain everything.NOTES1. After his debacle with the birthday problem on Feb. 6, Johnny Carson tried two moreexperiments with his studio audience on the following nights. Take a look at these clipsfromFeb. 7andFeb. 8— they contain instructive errors of their own, as well as pricelesscomedic moments.2. If you’d like to watch the rest of these or any of the other surviving episodes of “TheTonight Show,” they can be viewed at theCarson archives,but first you’ll need to registerfor a free account. You can also find dozens of immortal television moments by pokingaround at theCarson Web site,reviewed in thisnews feature.3. For those too young to remember who Johnny Carson was, see the American Mastersdocumentary “JohnnyCarson: King of Late Night.”As thePBS Web sitenotes, Johnnywas “seen by more people on more occasions than anyone else in American history. Overthe course of 30 years, 4,531 episodes and 23,000 guests, he became a fixture of nationallife and a part of the zeitgeist. In a 2007 TV Land/EntertainmentWeeklypoll, Americansvoted Carson the greatest icon in the history of television.”4. The quoted excerpt from “Innumeracy” appears on page 36 of J. A. Paulos,“Innumeracy” (Hill and Wang, 1989). You can find other versions of the same story (manyof which have mutated as in the children’s game of “telephone”) by searching the Web orGoogle Books for “Johnny Carson birthday problem.”5. The solution of thebirthday problemgiven above contains three standard simplifyingassumptions. It assumes there are 365 days in a year (which ignores leap years); allbirthdays are equally likely (which ignores daily andseasonal variations in birth rates);and all birthdays are chosen independently (which ignores the possibility of twins ortriplets or any other non-random connections that tie people’s birthdays together).In reality, birth rates are not uniform throughout the year. Furthermore, the pattern ofvariation is different in different parts of the world. In the United States, for example,daily birth rates rise in the summer and fall in the winter, ranging from about 7 percentabove average in September to 5 percent below average in January. Yet the answer to thebirthday problem remains 23 even after these seasonal variations are taken into account,as shown in T. S. Nunnikhoven, “Abirthday problem solution for nonuniform birthfrequencies,”The American Statistician, Vol. 46, No. 4 (Nov., 1992), pp. 270–274 andfurther discussed in M. C. Borja and J. Haigh, “Thebirthday problem,”Significance, Vol.4, No. 3 (September 2007), pp. 124–127, and the references therein.http://www.readability.com/articles/l2vscn6p5/6 [ Pobierz całość w formacie PDF ]