Understanding the quirks

Solutions

Question 1

Question 1 is a variant on what has come to be known as the Linda-the-bank-teller problem. To disguise this somewhat (because the actual problem is now quite well known) the question asks about Shelia rather than Linda but the problem is otherwise identical to the original. The problem reflects on something called the conjunction fallacy (alternatively, the conjunction error). Let’s start with a basic fact – the probability of two events occurring together cannot be greater than the probability of either event occurring alone.

For instance, it can rain and it can thunder, and, of course, it can rain and thunder at same time. But thundery rain cannot be more probable than either rain or thunder alone.

If you really want to dig deeper then you can delve into conditional probabilities, but let’s here just stick to cognitive psychology. The classic finding is that the vast majority of people rate:

Shelia is a bank teller who is active in the feminist movement.

More probable than:

Shelia is a bank teller.

Hence the conjunction fallacy – people respond as though the conjunction of being a bank teller and being active in the feminist movement is more probable than being just a bank teller and this is just plain wrong. People are reasoning fallaciously (falsely).

So what on earth is going on? Well rather than give the game away just yet, let’s move on – all will be revealed as the material unfolds!

Question 2

Question 2 taps into something quite different, and people’s responses here reveal a rather basic failure to appreciate another fact about the world. This reflects back to Week 1 where we discussed the nature of scientific investigations such that the more often we replicate a finding then the more confident we can be in how robust the finding is. A robust finding is something we take to be something that cannot be a chance event.

Question 2 taps into a related fact. Let’s randomly sample 1000 people and record their sex. Let’s repeat this 100 times. Across the 100 sets of recordings 50% male/female breakdown will shine through and moreover, all of the 100 recordings will hover around the 50% figure.

Let’s now do this but instead of repeatedly sampling 1000 people we repeatedly sample 2 (yes 100 samples of two people). Across these samples, 25 will comprise two males, 25 will comprise two females and 50 will comprise one male and one female. So ¼ of the records will contain more males than females and half of the recordings depart from our 50% gender breakdown.

In other words, the recordings made from small samples are far more likely to stray markedly from the 50% breakdown than are the recordings made from large samples. Therefore in answer to the Question 2, it is far more likely to find days on which more boys than girls are born in the smaller than the larger hospital.

This kind of basic understanding about the nature of the world appears not to feature heavily in our reasoning about it!

The original finding is conveyed below:

The larger hospital (21) The smaller hospital (21) About the same (53)

The numbers in brackets show the number of students who chose the particular answer and, overwhelmingly, the favoured option was about “About the same”.

It is tempting to state, “most of us are not trained scientists” but it would not be at all surprising to find that many scientists are also tripped up by this problem!