Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 11 / d

Here “absent” denotes absence at both observed lectures. The assumed independence makes its conditional probability

For attendance probability $9/10$, absence on both independent observations has conditional probability

The law of total probability over the two student groups gives

The prior probability is $1/10$. Using Bayes' theorem, the posterior probability after two absences isThus the probability rises from 10% to about 73.5% within the assumed attendance model. Two absences are 25 times as likely under the lazy model, overcoming the prior odds of $1:9$; they still do not make the classification certain.