  The detection loophole problem arises when quantum devices fail to provide an
output for some of the experimental runs. These failures allow for the
possibility of a local hidden-variable description of the resulting statistics;
even if the correlations obtained from successful runs appear non-local. This
is of particular significance in device-independent quantum cryptography, where
verifiable non-locality is a necessary requirement for the security proof. For
every scenario characterised by the amount of devices, along with the number of
inputs and outputs available to them, there is a detection threshold - if the
efficiency of the device falls below this no verification is possible. In this
work I present an intuitive local hidden-variable construction for all
no-signalling distributions with two parties and binary outcomes. This provides
a lower bound on the detection threshold for quantum measurements in the same
scenario tighter than known previously. When both parties have the same number
of inputs into their device, this construction is shown optimal for small input
numbers. I finish with some interesting conjectures.
