The Watson Selection Task – a logical puzzle

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The Watson Selection Task is a logical problem designed to show how bad we are at making logical decisions.  Watson first used it in 1968 – and found that only 10% of the population would get the correct answer.  Indeed around 65% of the population make the same error.  Here is the task:

The participants were given the following instructions:

Here is a rule: “every card that has a D on one side has a 3 on the other.” Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.  Each card has a number on one side and a letter on the other.

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Give yourself a couple of minutes to work out what you think the answer is – and then highlight the space below where the answer is written in white text.

The correct answer is to pick the D card and the 7 card

This result is normally quite unexpected – but it highlights one of the logical fallacies that we often fall into:

A implies B does not mean that B implies A

All cats have 4 legs (Cats = A, legs = B, A implies B)
All 4 legged animals are cats (B implies A)

We can see that here we would make a logical error if we concluded that all 4 legged animals were cats.

In the logic puzzle we need to turn over only 2 cards, D and 7.  This is surprising because most people will also say that you need to turn over card with a 3.  First we need to be clear about what we are trying to do:  We want to find evidence that the rule we are given is false.

If we turn over the D and find a number other than 3, we have evidence that the rule is false – therefore we need to turn over D.

If we turn over the 7 and find a D on the other side, we have evidence that the rule is false – therefore we need to turn over the 7.

But what about the 3?  If we turn over the 3 and find a D then we have no evidence that the rule is false (which is what we are looking for).  If we turn over the 3 and find another letter then this also gives us no evidence that the rule is false.  After all our rule says that all Ds have 3s on the other side, but it doesn’t say that all 3s have Ds on the other side.

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Are mathematicians better at this puzzle than historians?

Given the importance of logical thought in mathematics, people have done studies to see if undergraduate students in maths perform better than humanities students on this task.  Here are the results:

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You can see that there is a significant difference between the groups.  Maths students correctly guessed the answer D7 29% of the time, but only 8% of history students did.  The maths university lecturers performed best – getting the answer right 43% of the time.

Making different mistakes

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You can also analyse the mistakes that students made- by only looking at the proportions of incorrect selections.  Here again are significant differences which show that the groups are thinking about the problem in different ways.  DK7 was chosen by around 1/5 of both maths students and lecturers, but by hardly any history students.

You can read about these results in much more depth in the following research paper Mathematicians and the Selection Task – where they also use Chi Squared testing for significance levels.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Essential Resources for IB Teachers


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If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

  1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
  2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
  3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

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I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.