Retaking exams alters their informativeness

If only those who fail are allowed to retake an exam and it is not reported whether a grade comes from the first exam or a retake, then the failers get an advantage. They get a grade that is the maximum of two attempts, while others only get one attempt.
A simple example has two types of exam takers: H and L, with equal proportions in the population. The type may reflect talent or preparation for exam. There are three grades: A, B, C. The probabilities for each type to receive a certain grade from any given attempt of the exam are for H, Pr(A|H)=0.3, Pr(B|H)=0.6, Pr(C|H)=0.1 and for L, Pr(A|L)=0.2, Pr(B|L)=0.1, Pr(C|L)=0.7. The H type is more likely to get better grades, but there is noise in the grade.
After the retake of the exam, the probabilities for H to end up with each grade are Pr*(A|H)=0.33, Pr*(B|H)=0.66 and Pr*(C|H)=0.01. For L,  Pr*(A|L)=0.34, Pr*(B|L)=0.17 and Pr*(C|L)=0.49. So the L type ends up with an A grade more frequently than H, due to retaking exams 70% of the time as opposed to H’s 10%.
If the observers of the grades are rational, they will infer by Bayes’ rule Pr(H|A)=33/67, Pr(H|B)=66/83 and Pr(H|C)=1/50.
It is probably to counter the advantage of retakers that some universities in the UK discount grades obtained from retaking exams ( In the University of Queensland, those who fail a course can take a supplementary exam, but the grade is distinguished on the transcript from the grade obtained on first try. Also, the maximum grade possible from taking a supplementary exam is one step above failure – the three highest grades cannot be obtained.

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