We suck at probability

posted in: General science, Philosophy | 2

Introduction:

You are probably bad at probability.  It’s okay.  Probability is a concept that our filthy monkey brains have not had to grapple with for very long.  This is because our brains evolved to err on the side of caution.  Our ancestors did not increase their reproductive success by calculating the actual odds of something being a hungry sabretooth tiger.  They lived longer and had more babies when they were generally terrified and reacted to everything as though it were a sabretooth tiger.

The problem with this inherent difficulty with probabilistic thinking is that it can lead to serious confusion.  Sometimes, this confusion even seems to be intentionally cultivated for the benefit of otherwise untenable arguments.  Obviously, this is bad for the honest exchange of ideas and you should be aware of the potential for mistakes.

What is probability?

Probability is the quantification of the likelihood that some event will happen.  In other words, it’s an attempt to measure the odds of an event.

Here are two really simple examples.  A coin has two sides.  If it is tossed it has a 1/2 (or a 50%) chance of landing on either of its sides.  A six-sided die has a 1/6 (or 16.67%) chance of landing on any of its sides.

But probability is a much more complicated notion than the above examples imply.  The birthday problem and the Monty Hall problem are great examples of how bad human intuition is at probabilistic thinking.

The birthday problem:

The birthday problem asks: how many random people does it take for there to be a 50% chance that two people share the same birthday?

Well, we can be sure that with 367 people there is a 100% chance that at least two of them share a birthday.  This is because there is only a maximum of 366 days in a (leap) year.  But would you believe that we reach a 50% chance of two people sharing a birthday with just 23 people?!  Crazy huh?  And with 70 people there is a 99.9% chance that two people will share a birthday!  If you don’t want to take my word for it, check out these explanations.  Or, you could make a bit of a scene on your next bus ride and see for yourself.

The Monty Hall problem:

The Monty Hall problem asks if your chances are improved by changing your initial guess after a set of potential guesses is reduced.  This problem takes a bit of setting up.

Imagine you are shown three identical doors (“A”, “B” and “C”).  Then you are told that behind two of them are ornery, poop-spattered goats.  But behind one of them is a friendly Pegasus that smells of rich mahogany (Figure 1).  Finally, you are asked to guess what door you think Pegasus is behind.

A majestic Pegusus and a goat with its head in another goat's butt
Figure 1: A regal, winged stead, or a shitty goat?

Let’s say you guessed door “A”.  Then, you are kindly shown what is behind door “C”.  It’s a stinking goat.  Now that you know Pegasus is behind either “A” or “B”, you are given the opportunity to change your initial guess.  So here’s the question: is it better to change your guess to door “B”?

Most people intuitively say that it makes no difference whether you change your guess or not.  But, in reality, at this point, if you switch to door “B” you have a 2/3 (66%) chance of picking the Pegasus.  This is because the revelation of the shitty goat behind door “C” added value to the unrevealed and un-guessed door (“B”).  Here is a more thorough explanation.

A common misuse of probability:

Okay; probability is counterintuitive.  So, now let’s examine one particular way that it can be misused to make a point.

You’ve likely come across examples of people trying to highlight the extraordinary nature of an event by calculating its probability.  This is often done for events that have already happened.  An example might be the odds of someone winning the lottery.  However, while the odds of winning the lottery are certainly very bad, the odds of someone winning are pretty good.  So for the winner to be described without that context is a big mistake.

Another more grandiose example is the claim that the existence of the universe, or the nature of its properties, is so improbable that it may as well be considered impossible.  This is often followed by the assertion that some supernatural entity must have at least helped the universe along.  A subtler way of expressing this is to claim that the universe must have been “designed.”  Here is a great, and sophisticated, example of this assertion made by William Lane Craig:

“Consider gravity, for example. The force of gravity is determined by the gravitational constant. If this constant varied by just one in 1060 parts, none of us would exist.”

Craig goes on to ask himself a question on how this aforementioned precision came to be (he also answers it):

“How about chance? Did we just get really, really, really, really lucky? No. The probabilities involved are so ridiculously remote as to put the fine-tuning well beyond the reach of chance.”

I would have thought that if we had gotten that lucky, I mean “really, really, really, really lucky,” that we would live in a world with gravity and without disease, rape and war, but that’s just me.  Anyhow, his own answer then leads him to assert that:

“Given the implausibility of physical necessity or chance, the best explanation for why the universe is fine-tuned for life may very well be it was designed that way.”

Of course, it seems like quite a large number might also be required to define the probability of such a designer actually existing, but this is not mentioned in the above argument.  Another flaw in this line of reasoning is that we have no knowledge of anything outside of our own universe.  Thus, we do not have the requisite information to assess the probability of this event.  Maybe universes like ours pop into existence all the time.  We just don’t know.  And not knowing that is alright—and it’s important to admit this.

Yet another problem with this is that calculating an event’s probability after it has happened can be a bit like throwing a dart at a wall and then drawing a bullseye around it.  Imagine painting some circles, holding the brush behind your back and then proclaiming: “Whoa, what are the chances that I would throw a bullseye on the first try!”

Here’s an illustration: a footprint on a sandy beach must compact some specific and extremely high number of sand grains.  These sand grains will also have displaced some staggeringly large number of water molecules.  Thus, the actual probability of any footprint disturbing some specific set of beach particles is extraordinarily high.  It’s probability might even be described as being “ridiculously remote.”  But we have to remember that some set of particles had to be impacted by the foot.  If there is nothing particularly unique about the specific particles that were affected, then calculating the odds of the footprint’s consequences is mental masturbation.  It’s akin to drawing a bullseye around a thrown dart.  This pointless calculation certainly shouldn’t have any bearing on honest discussions of whether or not the Earth is capable of moving sand and water to the precise spot that a foot happened to land.

Conclusion:

Humans suck at thinking about probabilities.  We have to work hard to get them right.  But what is also important is to work hard to make sure others don’t use bogus odds to mislead us.

Jared Peters

Jared Peters

Jared Peters, PhD, is a geoscientist who specialises in marine sedimentology, marine palaeoglaciology and climate change.
Jared Peters
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Jared Peters, PhD, is a geoscientist who specialises in marine sedimentology, marine palaeoglaciology and climate change.