A shop window is broken and part of the displayed goods is taken. A witness claims that the thief was an Arab. When investigators propose similar scenes in similar light conditions, distance, … the witness correctly identifies the race (Arab, non-Arab) of the thief in 75% of cases.  Is the testimony reliable if in the locality considered 12% of the thefts are the work of Arab thieves?  [to answer, calculate the probability that the thief is actually an Arab]

I have to evaluate Pr("to be an Arab thief" | "to be a thief taken for Arab"), that is Pr("to be a thief" AND "to be Arab") / Pr("to be a thief taken for Arab").  Let's use a graph to illustrate the situation and do the math.

                 identif. OK  75% ---- 12%·75%
     Arab          (Arab)    /
    thief  12% --------------
          /      identif. KO \
         /        (not Arab)  25% ----
 --------
         \       identif. OK  75% ----
not Arab  \       (not Arab) /
 thief   88% --------------
                 identif. KO \
                   (Arab)     25% ---- 88%·25%

Pr("to be a thief" AND "to be Arab") = 12%·75%.
Pr("to be a thief taken for Arab") = 12%·75% + 88%·25%
Pr("to be an Arab thief" | "to be a thief taken for Arab") = 12·75/(12·75+88·25) = 9/31 = 0. 2903225… = 29.0%.  It's a pretty low probability!
Note the importance of realizing how easy it is to deceive yourself into making probabilistic assessments.  It is perhaps more important to be aware of the errors we may meet (and which television, judges, law enforcement officers, ... often meet), and, in face of situations of this kind, possibly look at a manual or ask someone who you know is competent, rather than knowing how to mechanically deal, only at school and for the necessary time, questions of this type.