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The Lazy Student Formula

How to minimise your revision


It's exam season, and for the lazy student it's time to figure out how little revision you can get away with without revealing huge gaps in your knowledge.  

A listener to Radio 4's More or Less sent in an interesting query about this, and I was asked to investigate the maths behind it.

Suppose you are taking an exam in History, and that there are 20 topics in the syllabus.  The end of year exam will have twelve questions, each on a different topic.  In the exam, the candidate is only required to answer three questions. 

How many topics do you need to revise in order to be 100% certain that you will have at least three questions that you can tackle in this exam?

It's a nice little puzzle, and you might want to work out the answer before reading on.

                      *                       *                        *                       *                       *

To figure out how many topics you need to revise, imagine the worst case scenario. In this mythical History exam, the maximum number of questions that could be on topics that you haven't revised is eight (= 20 - 12).  So if you revise eight subjects, plus an additional three - i.e. 11 topics in total - then you are guaranteed to have at least three questions to choose from in the exam.  

There's a simple formula to work out the minimum revision level for any exam of this type.  Suppose there are:

       T topics in the syllabus

       Q questions in the exam (each on a different topic)

       A questions must be answered

To be 100% certain that at least A questions will come up in topics that the student has revised, the number of topics revised, R, is :

               R >= T – Q + A

Just to confirm it works - if there are 20 topics and 12 questions of which 3 must be answered, R = 20-12+3 = 11.     

But can the lazy student save even more effort?  For the 20/12/3 exam above, then if the student only revises ten topics – instead of eleven – it is still almost certain that they will get at least three questions on revised topics. The chances work out to be 99.96%* .  So although revising only ten topics is a risk, it is a very tiny one.

And it gets better for the lazy student.  They can still be 95% confident of getting three questions even if they only revise seven topics out of 20. 

However, the risk climbs rapidly once the revision drops below that level.  If they revise only four topics, the chance of getting three questions falls to less than 50-50. 

Here's a table of how the risk grows as revision breadth is reduced:

No. of topics revised            Probability of >= 3 questions coming up*

              9                                                       99.7%

              8                                                       98.5%

              7                                                       94.8%

              6                                                       86.3%

              5                                                       70.4% (now entering high risk territory)

              4                                                       46.6%

              3                                                       19.3%

Is there a simple back of envelope rule to work out how much little the lazy student needs to revise in order to be able to have a good stab at the exam?  Yes there is - but remember that it is only a crude guide and not a definitive rule.

If you want at least a 90% chance of questions coming up on topics you revised, the number of topics, R, that you should revise in depth is very roughly given by:

         R =    T x A   + 1      (if Q>A and if A>2)  **

                      Q

Does it make sense to follow this rule?  If you are a gambler with not enough time to spend on studying then maybe.  But remember, the formula assumes that if a question comes up on a topic you have studied, you will be able to tackle it successfully.  It also assumes that no questions make links between topics. Real exams are rarely like that.  So it generally pays to have at least a couple of spare topics up your sleeve just in case.

So what's my tip?  Well of course, I advise that any student should study the whole syllabus in depth so you are fully equipped to tackle anything that comes up in the exam.  Just like I always did. (Not).

* One way to work this out is to calculate the chance that exactly two, one or zero topics will come up in the exam, and subtract this from 100% to get the chance that at least 3 topics come up.  Note that if you revise ten topics, the chance of exactly 1 or 0 topics coming up in the exam is zero.

**  Note that this back of envelope formula doesn't work if all questons are compulsory.  When Q=A, the formula suggests you need to swat up on T+1 topics, which means going beyond the syllabus!  And the formula becomes much less reliable when you are only expected to answer 1 or 2 questions out of not many in the exam.