## Labels

### A problem in probability

This is from an old Cambridge STEP paper from 1998.
The diagnostic test AL has a probability 0.9 of giving a positive result when applied to a
person suffering from the rare disease mathematitis. It also has a probability 1/11 of giving a
false positive result when applied to a non-sufferer. It is known that only 1% of the population
suffer from the disease. Given that the test AL is positive when applied to Frankie, who is
chosen at random from the population, what is the probability that Frankie is a sufferer?
In an attempt to identify sufferers more accurately, a second diagnostic test STEP is given
to those for whom the test AL gave a positive result. The probablility of STEP giving a
positive result on a sufferer is 0.9, and the probability that it gives a false positive result on a
non-sufferer is p. Half of those for whom AL was positive and on whom STEP then also gives
a positive result are sufferers. Find p.
The first part is straightforward: I get p(M|A)=1/11, using M for 'has mathematitis', A for 'passes the AL test', and S 'for passes the STEP test'.  The second part seems to me to be underspecified. It's asking for p(S|M'). I interpret
Half of those for whom AL was positive and on whom STEP then also gives
a positive result are sufferers
to mean that p(M|A&S)=1/2.  But I can't see how that gives any handle on p(S|M'). Indeed p(S|M') could take many values consistent with the data.

What do you think?