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Large numbers

What are the chances that…

A while back, I had a story mentioning a Cornell law professor named Michael C. Dorf. Part of the discussion revolved around correctly identifying the person from his name. I mean maybe there is more than one Michael Dorf, but more than one Michael C. Dorf? And even if there were multiple Michael C. Dorfs, surely there wouldn’t be two attorneys with that name. And in the hugely unlikely event that two attorneys share that name, it’s unthinkable that they could be both linked to Barack Obama.

Only, there are two of them. One is the Cornell law professor that wrote a paper on presidential eligibility, and the other is a Chicago attorney who actually represented Barack Obama.

I wrote the preceding as if this were an amazing coincidence, but I don’t think it is all that amazing. I mean Dorf is an unusual surname: it ranks number 35,938 in the 1990 census (US Census tabulation). Michael, however, is quite common and C is a common initial. There are lots of attorneys too, 1,225,452 according to the American Bar Association. What perhaps does make this instance really unusual is the connection to Obama, but even that connection is tenuous. The Cornell professor really isn’t connected to Obama except that he wrote an article about presidential eligibility, specifically the possibility of a president achieving a third term by being elected vice president after having served as President. The Chicago attorney’s association is more direct, but back in the past, when Obama was a state senator.

There are three errors of thinking we make in spotting remarkable coincidences (or are they?). The first is to fail to realize that when we talk about the population of the United States, some 300 million people, that a lot of infrequent coincidences are statistically likely. I remember doing quality assurance on a large statewide database, checking for duplicates, and being struck by the number of people born on the same day with the same name,  and this wasn’t even a large state.

The second error is to fail to consider how encompassing the criteria are, and whether the criteria are being manipulated to include a coincidence. In the Dorf example, the category of connection to Obama was expanded, and if that hadn’t worked out, perhaps the criteria would have been “lawyers from Illinois” or “Democrats” or “went to the same law school” or something else. It is one thing to ask “what is the chance that …?” before the fact and quite another to ask “what is the chance that we can find some connection given all the possible connections we could look for?” after the fact.

The third error is to look at any particular unusual event and to assign significance to it. Say that we conclude accurately that we are looking at a one in ten thousand event. But if there are a million people spending hundreds of millions of hours searching for unusual events linked to Barack Obama, chances are that quite a few unlikely (on their own) events will be found.

When a large number of unlikely events is presented in a list, they appear extremely unlikely to have all happened, but such lists are not given in the context of the other list, many orders of magnitude larger, of things that are not unusual at all.

We humans are well-adapted to recognize and assign significance to unusual occurrences. We are not, however, well-adapted to dealing with large numbers and the wealth of information available on the Internet. What looks unusual may not be.

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13 Responses to Large numbers

  1. avatar
    Slartibartfast December 1, 2013 at 1:48 pm #

    Not understanding that when the sample space is incredibly large then unlikely events are expected to happen is a common feature of conspiracy theories. Creationists like to use this misunderstanding to imply that abiogenesis is so unlikely as to be impossible.

  2. avatar
    Thinker December 1, 2013 at 4:06 pm #

    Back in the pre-internet days, I had a part-time job shelving books in a university library. Students would often come up to me to ask about a particular book or journal that was not on the shelf where it should be. I cannot tell you how many people simply refused to accept that someone else might be reading the exact book or article that they wanted to read. I think their reasoning was that, with hundreds of thousands of books and journals in the library, what are the chances that two people would want to read the same publication at the same time?

    Well, pretty high when you want the most popular publications the library carries.

  3. avatar
    Rickey December 1, 2013 at 5:26 pm #

    Doc’s link to the census tabulation got me thinking – how many people named Bounel are listed in the 1990 census?

    The answer: zero.

    This doesn’t mean that there were no people named Bounel in the U.S. in 1990, because the census doesn’t find everyone. But it confirms what we previously believed, namely that “Bounel” is an extremely rare name, close to non-existent.

  4. avatar
    jayHG December 1, 2013 at 9:40 pm #

    Coincidence…..I think not!! Birthers need to get more creatie. Just apply six degrees of separation. They’ll be able to come with a connection between President Obama and say me…..or Jeffrey Dahmer…..or….

  5. avatar
    JRC December 2, 2013 at 1:01 am #

    This kind of thing is used as propaganda often. Lord Mockingbird used it with the birth certificate. But watching the JFK assassination program, they explain it well. The gist is that if someone wore a blindfold and shot into a building 3 or more time, and someone can’t duplicate the randomness by being blindfolded and hit the same exact spots an the same amount of time then is there a question about how impossible those first shots were made. Even though those shots are just what they are, and completely possible, some will argue that they were impossible.

  6. avatar
    The Magic M December 2, 2013 at 5:59 am #

    JRC: The gist is that if someone wore a blindfold and shot into a building 3 or more time, and someone can’t duplicate the randomness by being blindfolded and hit the same exact spots an the same amount of time then is there a question about how impossible those first shots were made. Even though those shots are just what they are, and completely possible

    Another angle to look at it is:
    Things that may seem highly unlikely are a pointer that there *may* be something worth investigating further, but they are never proof in itself that things aren’t as they seem.
    Case in point, just because Lee Harvey Oswald made three “astonishingly precise” shots doesn’t mean he didn’t make them by pure luck. A good investigator would take such an “improbable” thing as starting point for looking deeper, but if nothing comes out of it, it means the improbable thing was just what it was – and not that some alien technology or magic (or a cover-up of epic proportions) was involved.

    Birthers are falling in a similar trap. “Inconsistencies” about Obama (life story, birth certificate), even if they were, arguendo, considered as such, don’t prove forgery/fraud/fould play. In fact, despite the “hundreds” of perceived “red flags”, no amount of digging has brought up anything substantial (like solid proof of foreign birth or loss of citizenship). Birthers OTOH act as if the “red flags” on their own were proof of anything.

  7. avatar
    Andrew Vrba, PmG December 2, 2013 at 9:21 am #

    “Large number” is Birther-speak for “More than three”.
    I.E. “A large number of people are going to be supporting the latest movement to oust Obama.”

  8. avatar
    Pastor Charmley December 2, 2013 at 3:29 pm #

    I am reminded of a common abuse of statistics, namely the claim that because the chances of an event happening, or a coincidence occurring, therefore the said event cannot have happened. Such statistics are not there to judge what HAS happened, they are there to tell us the probability of the event happening in the future. One cannot use statistics to deny reality. After all, every time an outsider wins a horse race, an improbable event happens.

  9. avatar
    Rickey December 2, 2013 at 3:44 pm #

    Pastor Charmley:
    I am reminded of a common abuse of statistics, namely the claim that because the chances of an event happening, or a coincidence occurring, therefore the said event cannot have happened. Such statistics are not there to judge what HAS happened, they are there to tell us the probability of the event happening in the future. One cannot use statistics to deny reality. After all, every time an outsider wins a horse race, an improbable event happens.

    Yes, they take something which is improbable and call it impossible.

    Last year I was with a group of people who were getting a private tour of Paramount Studios. We were broken up into small groups and each group had a member of the studio’s public relations department take us around. While we are having lunch one member of our group asked the PR person where she came from. She responded, “Westchester County, New York.” This intrigued me because I grew up in Westchester, so I asked her where she went to high school. It turned out that she went to the same high school as me, although roughly 40 years later. It is a relatively small high school, with a graduating class of less than 150. So what are the odds that I would travel to Hollywood and meet a total stranger who went to the same small high school as me? Astronomical, I would think, but it happened.

  10. avatar
    Dr. Conspiracy December 2, 2013 at 4:13 pm #

    The great story of coincidence for me was the one in the book, Lady Luck: the theory of probability by Warren Weaver. His Chapter 13 is titled ” Rare Events, Coincidences and Surprising Occurrences.”

    It was the story of George D. Bryson who made an unplanned stopover over in Louisville, KY on the way to New York on a business trip. He checked into the Brown Hotel and for a lark, asked if there was any mail for him. There was, addressed to George D. Bryson, Room #307 (his room). The previous occupant of the room was also named George D. Bryson.

    I have my own stories. How many times have you filled out an entry form and dropped it into a box at some merchant? How often do you win? I may have filled in a dozen such cards in my whole life. Among my winnings were TWO bicycles–one a Dr. Pepper bike at Wendy’s and one a Sorento Cheese bike at a supermarket. What is the chance that someone would win two bicycles?

    In Ukraine, we met maybe four American strangers on the street. One of them went to the same college as my wife.

    Rickey: So what are the odds that I would travel to Hollywood and meet a total stranger who went to the same small high school as me? Astronomical, I would think, but it happened.

  11. avatar
    Rickey December 3, 2013 at 5:06 pm #

    Dr. Conspiracy:

    I have my own stories. How many times have you filled out an entry form and dropped it into a box at some merchant? How often do you win? I may have filled in a dozen such cards in my whole life. Among my winnings were TWO bicycles–one a Dr. Pepper bike at Wendy’s and one a Sorento Cheese bike at a supermarket. What is the chance that someone would win two bicycles?

    The odds of winning the lottery are miniscule, but eventually somebody wins. I have friends who play the lottery and they always tell me “Somebody has to win.” My response is “Yes, but it won’t be you.” I’ve never been wrong! Of course, I may be jeopardizing my prospects of sharing in future good fortune by being a smart ass.

    In 1971 Off Track Betting was new in New York. Before the Belmont Stakes that year at work we came with the idea of having a betting pool where each person would chip in $2 and we then bought a $2 win ticket on each of the horses in the race. Then we put the tickets into a hat and each person drew one. I ended up with a longshot named Pass Catcher, who went off at odds of 34-1. Pass Catcher took the lead with a 1/4 mile to go and held on to win by half a length. My winning ticket paid $71. Up until then only one horse had won the Belmont Stakes at longer odds than Pass Catcher. When you factor in that the odds against me picking the ticket for Pass Catcher were 12-1, perhaps you can calculate what the eventual odds against me winning the pool were.

  12. avatar
    predicto December 3, 2013 at 8:34 pm #

    When I was in college, I met a fellow in the business school who had my reverse name.

    I don’t want to give my real name on the internet, so lets assume my name is something like Rufus Michael Harrison. His name was Harrison Michael Rufus.

    What are the odds of that happening with my particular name? Astronomical, of course. But the odds that something like that never happens with ANYONE’s name are even more astronomical. I just happened to me, not to that other guy. Probability is fun.

  13. avatar
    Keith December 3, 2013 at 8:55 pm #

    predicto: When I was in college, I met a fellow in the business school who had my reverse name.

    I don’t believe you. I mean, come on, somebody had a name ‘Otciderp’? How could you even pronounce that? 😎