One of the interesting things I came across as an undergrad was research in psychology on when and why natural human reasoning abilities work against, rather than for us, or how we are not inclined to think as rationally as we think we think (yes, the repetition of “think” was deliberate). So I thought I’d give some examples from the research into human reasoning and its irrationality.
The first example I like runs something like the following:
Imagine an individual. We’ll call this individual Alice. Alice is a native U.S. citizen and current resident between 25 and 35 years of age. She’s loves books. She is rather shy, and tends to prefer quiet and calm. She is also very organized. Though shy, she does enjoy assisting others with when it involves something she is passionate about.
Given this profile, is it more likely that Alice is a librarian or a salesperson?
The study which asked this kind of question found people usually answered it incorrectly. “Common sense” here tends to make people focus on the wrong things. The details about Alice tend to make one think “librarian” because the profile fits for what one might imagine a librarian to be. However, this is actually a rather straightforward probability (and set theory) question. It’s asking about likelihood. The details about Alice are fairly vague. However, in the U.S., the number of librarians is tiny compared to the number of salespeople. The details aren’t enough to overcome the massive disparity. It’s much more likely that Alice is a salesperson.
This example I use when teaching/tutoring high school and college students. I stole it.
Imagine a “fair coin” is tossed a number of times (i.e. one that is equally likely to land either heads or tails but will not land on its side). Let’s say H is heads and T is tails. Given, say, over a dozen tosses, we would expect a bunch of heads and a bunch of tails. We probably won’t get a perfect division, but we would expect something like the following:
What we would not expect, and rightly so, is to get all heads or all tails, like so:
In other words,the first coin toss above is far more likely to occur than coin toss #two. Right?
While people almost always responded that the first coin toss is more likely, this isn’t the case. Here again “common sense” fails. It tells us that something like the first coin toss is more likely, and it also tells us that the second coin toss is clearly unlikely. However, what (it appears) our natural reasoning ability tends to miss is that both coin tosses represent exact sequences of independent probabilities, all 1/2. In other words, the probability of either occuring is equal. The question wasn’t whether something which looks more like coin toss 1 would be more likely than #2, but whether the exact sequence of heads and tails in coin toss 1 would be more likely.
You go to the doctor’s for a regular annual check-up/exam. During your exam, the doctor runs a bunch of tests for various diseases (she knows you’ve been up to no good and expects the worse). A new strain of a sexually transmitted virus was recently reported in the area you live in. Luckily, however, your doctor informs you that a new test has beed developed to detect whether an individual has the strain using a fairly non-invasive procedure, and after extensive studies, for 99.9% of those tested who had the strain, the test showed they did. Given this (and your recent series of one night stands with strangers who robbed you during your sleep), isn’t the test a good idea? Or, to put it another way, given the accuracy, isn’t it a promising test?
Well, no. Not based on the information given. We’re missing a bit of crucial information: namely, false positives. What if the test is accurate 99.9% of the time when someone has the strain because it almost always says that a person has the strain? If the percent of false positives is, say, 80%, then it’s a terrible test.
The general idea of this one is taken (stolen) from a paper in the edited volume On Conditionals (or perhaps On Conditionals Again, I can’t be certain which). The author uses a clever story to illustrate the problem of relying on your courses in logic when making decisions, when common sense would serve you better.
A University student was out for a stroll in the city. We’ll call him James (his full name is James James Morrison Morrison Weatherby George Dupree, and at a young age his mother, despite James’ instructions, went to a certain area of the town she oughtdn’t to have and has been missing ever since). Unbeknownst to James, a local protest rally has turned into a riot, Making matters worse, the local police are out arresting people left and right in order to stop the destruction which happened after the last protest rally some weeks ago.
James just happens to walk into the area in which the chaos is occuring. Although he was initially curious to find the source of all the tumult he heard, now that he realizes what’s happening he decides the best course of action would be to leave quickly. As he is turning around, however, he sees a rock on the ground. But this is no ordinary rock (and James would know, as he’s been collecting rocks since he was three, initially to cope with the loss of his mother). So he picks it up. Unfortunately, it is at that moment that a police officer sees him. Thinking that James is about to throw the rock into the window of a nearby building or car, the officer arrests James.
A few days later, James finds himself in front of the Judge. He has already explained that he picked up the rock because embedded in it was a rather large specimen of a type of quartz not common to the area. The police officer has likewise given his testimony. The judge doesn’t feel there is any evidence, so he is inclined to rule not guilty. But he realizes (being the clever individual he is) that James never actually said he wasn’t going to throw the rock. So before he pronounces James innocent, he asks “If you weren’t arrested, then would you have thrown the rock through a window?”
James is a philosophy student, and having read Frege’s Begriffsschrift, the Principia Mathematica, and several other books which all contain systems of propositional and predicate logic, he knows exactly how to answer. After all, the judge has asked a conditional-an “if” question. As any intro to logic student would know, thinks James, a conditional, If a then b, only comes out “false” under one condition (he briefly draws a truth table just to double check; this isn’t the time to make mistakes). If the antecedent is true, the “if” part, and the consequent is false, the “then” part, the conditional is false. Otherwise, the conditional is true.
James realizes that the antecedent here is false. He was in fact arrested. Therefore, he reasons, his committment to logic allows only one answer, as the truth value of the whole conditional is clearly “true.” So he says “yes.” And is given a death sentence with the possibility of parole given good behavior after the execution.