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In the blogosphere, we often see arguments like this:

The odds of [some event] happening by random chance are incredibly low.  Therefore [event] was rigged or influenced to happen that way.

I've seen this argument applied to variations in voting patterns, timelines of terror attacks, the timing of bad news, and coincidences in general.  People are prone to suspect an evil actor behind unwanted events, and a mathematical argument seems to confirm our suspicions.  It's a very compelling argument.

It is also bogus, as you can observe any time you play cards.  Shuffle a deck of cards and look at the outcome:  the probability of that shuffle happening by random chance is 1 in 80658175170943878571660636856403766975289505440883277824000000000000.  Obviously this was rigged by a conspiracy while you weren't looking!

This is a common logical fallacy known as the confusion of the inverse.  I explain below.

The fallacy centers around the ambiguity of this question:

What is the probability that event X happened by chance?

This question has two confusable meanings:

  1. Under random chance, what is the probability of X happening?

  2. If X happens, what is the probability that random chance is the cause?

To write this conditionally, question (1) asks for Pr[X|Chance] and question (2) asks for Pr[Chance|X].  In probability, a "|" means "given that"; or to put it another way, everything to the left of the bar is the thing you are wondering about, and everything to the right of the bar is stuff you either know, or assume to be true.
These expressions are not the same:  they are inverses of one another, and have very different meanings and often very different values.  The conspiratorial blogger should be making a decision based on (2), but almost always people mistakenly compute (1), yielding an impressively tiny number that doesn't really mean anything, but which can sway an audience of laypeople.

Look everyone!  Little tiny numbers!

But this fallacy is bigger than simply using the wrong formula.  It also employs pseudoscience and argument from incredulity.  First the incredulity:  you're supposed to believe that an event couldn't have happened because its probability is so impressively tiny.  What is missing is the context:  incredibly tiny is in fact perfectly normal.

In reality, most anything that ever happens by chance has probability well nigh 0.  You spill some salt, and the scattering of grains takes on one of inexhaustibly many different outcomes, each with infinitesimal probability.  (1) is small!  And yet that incredibly improbable outcome did happen by chance, without any reason to suspect the unseen aid of space lasers.  
Even more mind-numbing is the fact that it will never happen again:  any individual card shuffle is so unlikely that, once seen, you can be guaranteed that it will never be seen again, for the rest of the lifetime of the universe, assuming the shuffling is fair.  There is something weirdly unintuitive about observing an event happen right in front of you, and immediately declaring that it can never happen.

This can be counter-intuitive because when we hear "probability 0," or even "odds of 1 in a million," we think "this can't have happened."  But that's not what probability means.  A low probability does not mean an event can't have happened; it does mean that if you predict that specific event to happen, in advance, then you are not going to be right.  

If that confuses you, just remember the old joke about the farmer who would shoot the side of his barn, and then paint a bullseye wherever he hit.  The probability of landing on the bullseye is only low if you declare the bullseye in advance.  That's what the number Pr[X] ultimately describes.

Now, multiply by the probability of aliens

Okay, the second fallacy behind the confusion of the inverse:  usually, the probability Pr[Chance|X] can't even be computed, because it doesn't have a well-defined value.  So making a mathematical argument is pseudo-scientific to begin with.

Often we can compute formula (1) (this is why so many people mistakenly use that value,) but to get formula (2) you need to know numbers that you can't possibly know.  We can see this using Bayes Rule:  the expressions (1) and (2) are related as follows:

Pr[Chance|X] = Pr[X|Chance](Pr[Chance]/Pr[X])

The right side contains parts that often have no meaningful value.  Pr[X] is the probability of X happening by any cause, from coincidence to conspiracies to space lasers.  You don't know that number.  And Pr[Chance] is the overall probability of no conspiracy, no space lasers.  If an election is going to happen tomorrow, what is the probability that it will be in some way rigged?  1? 0.1? 0.001?  How do you know?  

How would we even get an estimate of that number?  By examining previous elections?  That would only tell us the odds of election-rigging happening and being caught.  And what elections do we count?  All the races in that same district?  There aren't enough elections, and election personnel and machines change too rapidly, to get any useful estimate of the odds of a conspiracy.  It's like trying to tell if a coin is fair by observing three coin flips.

It is possible to compute this type of number in very carefully designed experimental circumstances, which is what scientists do.  You compare results of drug XYZZY versus the results of a placebo, and you design the experiment so that you know the precise numbers of each case.  Likewise, in engineering, we can use these formulas because we know all the probabilities of everything---having built everything.  If you want to decide if a received signal represents a 0 or a 1, you know Pr[0] and Pr[1], because you built the transmitter.  But none of this applies if you try to analyze, after the fact, an event that happened in the wild.

Adjust your brain resolution

There's one more mistake behind the fallacy of the inverse:  the idea that we can draw a box around an event and determine the odds.  The odds of what?  Where do you draw the box?  How much detail do you include?  

For example:  I roll six dice and get 3, 4, 1, 5, 6 ,2.  What are the odds?  One in 46656?  Well, maybe if you only look at the numbers on top.  Suppose you consider the exact positions where the dice fell, or their orientation---what are the odds of them landing like that?  Much smaller, certainly.

When people examine a real-world event and try to compute the odds of it happening, they have to choose what detail to include and what not to include.  This event resolution can make the probability much smaller, much larger, and that much more meaningless as a number.

So now that you know how it's't do it

I hope this diary will innoculate your brain against a common mathematical misconception.  If ever you see someone arguing over whether the Governor's memo was intentional and they bust out the odds, remember that those odds are often meaningless, and any argument based on them is largely pseudo-quantitative.

I guess the moral of this diary is this:  mathematics is a tool for evil, and if any of us try to convince you of something using mathematics, you should assume the opposite is true.  

Ha ha, just kidding, the preceding sentence is false.  Happy Wednesday.

Originally posted to Caj on Wed Dec 09, 2009 at 10:24 AM PST.

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Comment Preferences

  •  Tip Jar (19+ / 0-)

    Linking to a news article is journalism in the same sense that putting a Big Mac on a paper plate is cooking.

    by Caj on Wed Dec 09, 2009 at 10:25:00 AM PST

  •  Cool (1+ / 0-)
    Recommended by:

    but I think you might want to repost those equations as images.

    •  I figured, screw it (3+ / 0-)
      Recommended by:
      pontechango, G2geek, XNeeOhCon

      I was advised to use MathML, but it didn't work.  Also it was horrible and painful; I will never do that again.

      Especially considering that probability is one of the few branches of math where most expressions can be written in plain old Roman letters, with brackets and a vertical pipe.

      I keep a running list of things that worked perfectly well before the tech sector tried to "improve" it by intruducing hilarious complexity and hundreds of new failure modes.  Like VHS movies, overhead projectors, telephones, and analog television.  I guess I'll add typesetting:  $$e^{i\pi}=-1$$ was far too complicated, and clearly they had to replace that with 20 lines of angle brackets that won't display.

      •  I don't understand why HTML doesn't just accept.. (1+ / 0-)
        Recommended by:

        ...simple TeX expressions using some simple HTML tag.

        How hard can it be to render them properly?

        illegal, n. A term used by descendents of European immigrants to refer to descendants of Indigenous Americans

        by ricardomath on Wed Dec 09, 2009 at 11:51:54 AM PST

        [ Parent ]

      •  telephones and stolen elections. (0+ / 0-)


        I'm a telephone switching systems engineer and I agree that the network has been broken.  Not long ago we had hardwired residential phones that always worked, and digital PBX office phones that always worked.  Today we have sexy cellphones that take pictures and send telegrams (text messages) but sound like dog doo and interfere with conversation.  And we have IP-based office phones that cost twice as much to install but are less functional and require frequent service to keep 'em running.   Bah humbug to all of that.    

        Hint:  When buying an office phone system, use digital handsets within each building, use IP-telephony for home office extensions and to connect between different branches of the company if any.  Lower cost, more reliable outcomes.

        Stolen elections:  

        The 2004 election outcomes were analyzed by independent statisticians in regard to one variable:  the reported instances where a voter found that Diebold electronic voting machines registered a vote for the opposite presidential candidate.  In other words, you touched the screen for candidate A, and the machine showed a vote for candidate B, and the error persisted despite trying to re-enter the correct choice, to the point where you called an election official into the voting booth to try to fix it.  

        What they found was that the reported vote-flips favored Bush over Kerry almost without exception (many reported flips toward Bush, almost no reported flips toward Kerry), to a degree that was millions to one against chance.  Conclusion:  either a systemic flaw in the machines, or deliberate tampering, was responsible for those outcomes.  

        This was not a case of the kinds of logical fallacies you cite.  There were two possible values for the variable: flip toward Bush and flip toward Kerry.  The null hypothesis was clear to begin with: that vote flips would be evenly distributed between candidates (equal probability of an outcome in either direction, like a coin flip).  The operationalization used information from county election officials rather than from partisan sources, so it should have picked up both types of flips (toward Bush and toward Kerry) equally well.  

        Note also, if you try to offset for tabulated vote totals, then you end up even further from the null hypothesis:  if for example 55% of the voters in a district voted for Bush, then you would expect 55% of the reported flips to be complaints from Bush voters that the machines flipped their votes from Bush to Kerry, rather than 50%.  Sticking with 50% effectively gives a higher weighting to each instance of a Bush-to-Kerry flip.  

        These results were not anecdotal, and they stood up to peer review.

        So at least for the 2004 Presidential race, we can say with reasonable certainty that the outcome was affected by a systemic flaw in the voting machines.  

        This does not give us "proof" that the voting machines were rigged, since the systemic flaw could have been caused by factors other than hacking.  However, in combination with circumstantial evidence such as the behavior of Diebold employees in making "last minute rush updates" to the machines in the field, and the partisan statements from the president of Diebold that he would "deliver" the election for Bush, we can reasonably say that the 2004 presidential election was stolen.    

        •  One nitpick (0+ / 0-)

          The null hypothesis was clear to begin with: that vote flips would be evenly distributed between candidates (equal probability of an outcome in either direction, like a coin flip).

          This would not be the null hypothesis for a capacitive touch-screen.  
          If a bunch of touch screens are defective or miscalibrated and the reading sticks to one spot, you would not expect the spots to be uniformly distributed over the screen, with equal likelihood of being stuck on a candidate at the top of the screen and a candidate further down.  I would not be surprised at all if the stuck areas tended to be at the edges rather than the middle, for example.

          But the fact that these reported errors persisted despite multiple attempts strongly suggest that they were not intentionally programmed misbehavior, unless the programmer's intention was to get caught rather than affect the election's outcome.

          A programmer could simply have the machine record the wrong vote without making anything unusual on screen.  Even if they were forced to display Bush on screen to record a Bush vote, it would not make sense to continually flip the vote after multiple attempts to select Kerry.

          That doesn't rule out intentional malfeasance completely; a broken machine could be the result of voter vandalism.

  •  is this a new series? (0+ / 0-)

    disclaimer: I oppose the escalation and any contrary discussion of said escalation is just that.

    by terrypinder on Wed Dec 09, 2009 at 10:31:44 AM PST

  •  Very nice synopsis. (2+ / 0-)
    Recommended by:
    Caj, gooners

    But the odds of any person who already regularly employs this fallacy actually reading it, understanding it, and changing = 52357390589253 to 1


    I wish I would have added this to this diary

    This also reminds me of the idiotic argument some IDers use against scientific theories of the development of life on Earth (one example was in Ben Stein's terrible movie "Expelled").

    "The first reaction of a progressive should be not to look at who is the target of hate, but to reject the hate first." -RandomActsOfReason

    by XNeeOhCon on Wed Dec 09, 2009 at 10:41:26 AM PST

  •  Also, source code for this diary (1+ / 0-)
    Recommended by:
    social democrat

    How do you get those big numbers?  On a Mac or any Unix-like operating system, open a terminal and use the bc command.

    Actually, it's a bit of a pain to use it directly; I invoke it from Tcl.  To get the numbers for this diary I opened a terminal, typed tclsh, and entered the following at the % prompt:

    % proc math e { join [join [eval exec echo \"$e\" | bc]] "" }
    % math 1+1
    % math 6^6
    % for {set f [set i 1]} {$i<=52} {incr i} {set f [math $i*$f]} <br>% set f

    •  Er... (1+ / 0-)
      Recommended by:
      social democrat

      That <br> should be a new line.

      I better not typeset anything else today; maybe on my next attempt I'll break an ankle.

      •  for windows users... (0+ / 0-)

        ...the equivalent in PowerShell would be:

        PS C:> 1+1
        PS C:> [Math]::Pow(6,6)
        PS C:>$f = 1; 1..52 |% { $f *= $_ }; $f

        Prison rape is not funny.

        by social democrat on Wed Dec 09, 2009 at 11:18:45 AM PST

        [ Parent ]

        •  Dueling banjos (0+ / 0-)

          That's nice, concise syntax.  Let me add that to the Tcl interpreter:

          % proc know what {
            if ![info complete $what] {error "incomplete command(s) $what"}
            proc unknown args $what\n[info body unknown]
          % know {if {![catch {math $args} res]} {return $res}}


          % 1+1
          % 2^60

          [seems to work.  Now the 1..52 part:]

          % know {
                if {[regexp {(.*)\.\.(.*)} [lindex $args 0] - a b]} {
                   set c [lindex $args 1]
                   upvar 1 ; for {set $a} {$<=$b} {incr _} {uplevel 1 $c} <br>          return

          % set f 1; 1..52 {set f [math $f*$_]}
          % puts $f

          Truly, Tcl is the Play-Doh of languages.

          •  Again with the weirdness (0+ / 0-)

            That <br> should be a newline, and there should be two underscores after the upvar 1.

            I guess it's a good sign if writing the code is easier than posting it.

            For more info on Tcl, check out the awesome Wikibook --- from which I got the know procedure.

            •  to avoid the <br> weirdness... (0+ / 0-)

              ...use <tt> rather than <blockquote> (or perhaps <tt> within <blockquote>); adding <br> for line breaks won't be necessary.

              As far as the ability to screw with the interpreter and so "import" new syntax, I gotta say that's pretty frickin' cool. Not sure how often I'd use it, but then I have been know to overload an operator from time to time just to get cleaner, more intuitive syntax.

              For instance, PowerShell would really benefit from having the exponentiation operator ^. I'm not sure I know how to get there from here, though.

              Prison rape is not funny.

              by social democrat on Thu Dec 10, 2009 at 05:34:52 PM PST

              [ Parent ]

  •  Very good diary for dealing with CT nt (0+ / 0-)
    •  see my posting "telephones and stolen elections." (0+ / 0-)

      2004 was stolen and that's not CT, it's a logical conclusion based on findings of independent statisticians analyzing reported vote-flips on Diebold voting machines.   Details in my posting.  

      And FYI, I routinely fight CT where I see it, most obviously re. 9/11.  

  •  Once is happenstance, (0+ / 0-)

    twice is coincidence.
    Three times is enemy action.

    "If all else fails... immortality can always be assured by spectacular error."

    by mydailydrunk on Wed Dec 09, 2009 at 11:11:47 AM PST

  •  Quote of the week. (2+ / 0-)
    Recommended by:
    Caj, G2geek

    There is something weirdly unintuitive about observing an event happen right in front of you, and immediately declaring that it can never happen.

    illegal, n. A term used by descendents of European immigrants to refer to descendants of Indigenous Americans

    by ricardomath on Wed Dec 09, 2009 at 11:38:58 AM PST

  •  What are the odds??? (2+ / 0-)
    Recommended by:
    Caj, G2geek

    I tell my friends that if you're dealt five cards, the odds are 100% that you'll get five cards. And the odds against getting those particular cards are approximately 3 million to 1. So the odds of getting a royal flush (in a particular suit) in five cards is 3 million to one, but the odds of getting 2 diamond, 5 heart, 6 heart, J spade, and Q diamond are also 3 million to one.

    But nobody ever looks at their hand and says, "Wow! What are the odds of getting 2, 5, 6, J, Q?"

    Listen, strange women lying in ponds distributing swords is no basis for a system of government.

    by Dbug on Wed Dec 09, 2009 at 12:14:17 PM PST

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