  Bell inequality violations can be used to certify private randomness for use
in cryptographic applications. In photonic Bell experiments, a large amount of
the data that is generated comes from no-detection events and presumably
contains little randomness. This raises the question as to whether randomness
can be extracted only from the smaller post-selected subset corresponding to
proper detection events, instead of from the entire set of data. This could in
principle be feasible without opening an analogue of the detection loophole as
long as the min-entropy of the post-selected data is evaluated by taking all
the information into account, including no-detection events. The possibility of
extracting randomness from a short string has a practical advantage, because it
reduces the computational time of the extraction.
  Here, we investigate the above idea in a simple scenario, where the devices
and the adversary behave according to i.i.d. strategies. We show that indeed
almost all the randomness is present in the pair of outcomes for which at least
one detection happened. We further show that in some cases applying a
pre-processing on the data can capture features that an analysis based on
global frequencies only misses, thus resulting in the certification of more
randomness. We then briefly consider non-i.i.d strategies and provide an
explicit example of such a strategy that is more powerful than any i.i.d. one
even in the asymptotic limit of infinitely many measurement rounds, something
that was not reported before in the context of Bell inequalities.
