Logical Fallacies 29: The Statistical Breakdown (or False Lottery)

Video here:
http://www.youtube.com/watch?v=DS95LCJYNwM&feature=watch_response

This time, I’ll be speaking less about elemental logic as is the norm in these videos, and more about another subject utterly dependent upon it: statistics.  I’m bringing this up because of a number of people I have found myself dealing with who can’t seem to differentiate between “improbable,” “impossible,” and “miraculous.”

            I once came across a greeting card that said on the front: “You’re one in a million.”  Inside, it said, “...which means that, if you’re in China, there are a thousand other people out there just like you.”  Do you see the joke?  China is a country with a population of more than 1.3 billion.  So if only one in every million Chinese has, say, an extra rib, that’s more than one thousand three hundred people.

            What if one tosses a coin?  What are the odds that it will be heads?  Effectively, they are one in two.  Not precisely, of course, but close enough for this illustration.  What are the odds that it will be tails?  The same.

            What are the odds that it will land on its side?  Well, to be completely honest, they are not nonexistent, but they are small enough that we can probably afford to ignore them for this illustration.

            So what if you toss it three times?  Well given the fact that it’s one chance in two each time, two times two times two is eight.  One chance in two times one chance in two times one chance in two equals one chance in eight.  Therefore, since the odds that it will be tails each time are one chance in two, the odds that it will be tails all three times (two to the third) are one chance in eight.

            What about the odds that it will be tails twice then heads in that order?  One chance in eight.  What about the odds of it being tails, then heads, then tails in that order?  The same.  Tails, then heads twice?  Heads, then tails twice?  Heads, tails, heads?  Heads, heads, tails?  The same.  Each possible outcome of a single toss is one in two.  You just can’t predict which one.  Therefore, each possible outcome of three tosses is one in eight.  You just can’t predict which one.  Therefore, any time you toss a coin three times, the outcome will be something that had one chance in eight of happening.  The “odds” part doesn’t mean that something highly unlikely shouldn’t happen.  It only means that there is no way to predict which unlikely outcome will happen.

            When you toss a coin three times, any possible combination of heads and tails has one chance in eight of occurring.  Since the odds each time are one in two, those odds of a particular outcome double every time you add another toss to the mix.  So any time you toss a coin four times, the odds of any particular combination of heads and tails are one in sixteen.

            So what if you toss a coin ten times?  Two to the tenth is 1024.  So suddenly, the odds become one in 1024.  So if you take a coin and plan to toss it ten times, you can predict in advance that something incredibly unlikely is going to occur.  In fact, if you toss a coin ten times, there is no way to prevent something incredibly unlikely from occurring.  You just can’t be sure in advance exactly what it will be.

            What are the odds of someone winning the lottery with the numbers 1, 2, 3, 4 and 5?  In fact, they are exactly the same as with any other five-number combination.

            Now let’s consider this in a different context.  If you look at a professional weather report, you will probably find a percentage indicating a chance of precipitation.  What does it mean if the report says something to the effect of a ten-percent chance?  Does this mean that we can be absolutely certain that it will rain?  No.  Does it mean that we can be absolutely certain that it won’t?  No.  It means that, every hundred times the weather is like this, only ten of those times, on average, will result in rain.  Therefore, rain is unlikely, but not impossible, which means that if it does rain, that’s not a reason to go ridiculing meteorologists or meteorology.  It just means that this is one of those ten times in a hundred.

            I’m bringing all of this up because I recently discussed belief with a girl who argued that, since she was once in a car accident in which all the windows survived intact, there must be a god, who by the way, must be an involved god who gives a damn about windows in a car accident, and furthermore, must be the one from the Bible, even though that one can’t be bothered to protect other people’s windows in car accidents.  Her entire argument here depends on just how uncommon this kind of phenomenon is.  “It is uncommon.  Therefore it must be miraculous.  Therefore it must be the hand of god.  Therefore there must be a god.”  (Non sequitur!)

            Every car accident is absolutely unique.  Every time the windows do shatter, the ways they shatter are utterly unique to that accident.  Therefore every car accident is something highly unlikely.

            Every time someone gets into a car accident, we would be well founded in pointing at the windows and saying, “What are the odds of them shattering in precisely that arrangement?  They’ve got to be pretty small.”  This argument could be made in every single car accident.  Does this justify calling every car accident a miracle?  Surely it can’t.  Such would make miracles impossible to prevent.

            If the odds of a particular event occurring are small, that means this justifies the prediction of a low frequency of occurrence.  The odds of a car coming through a car accident with its windows intact are remote, so we are justified in predicting that it won’t happen that often.  We are definitely not justified in predicting that it will never happen unless in the event of a miraculous intervention, but completely justified in predicting that its occurrence will not be frequent.  Lo and behold, it doesn’t happen that often.  Its occurrence is not frequent.  Our prediction is borne out.

            Ah, but wait a minute.  There’s another wrinkle to this.  You see, we now live in an age of mass media.  I would call this a good thing overall, but we as a society need to understand the ramifications of this.  What if the odds of a particular event are one chance in a million?  That means that, on average, every million times it is possible for that event to take place, it will take place one of those times.  One just can’t be certain which?

            What if the odds of that something happening to a person on any given day are one in a million?  Well, that means that, on average, every day, that event is going to happen to more than 300 people in the United States alone.  We can predict this much, but we can’t predict which 300.

            Now, in an age of mass media, this means that, when that something happens to those 300 people, we’re all going to hear about it, and unfortunately, a lot of us are going to respond by saying, “Whoa!  What are the odds?”

            So I guess, logically speaking, one could call this a statistical breakdown.