  The standard formulation of quantum theory assumes that events are ordered is
a background global causal structure. Recently in
Ref.[$\href{http://www.nature.com/ncomms/journal/v3/n10/full/ncomms2076.html}{Nat.
Commun. {\bf3}, 1092 (2012)}$], the authors have developed a new formalism,
namely, the \emph{process matrix} formalism, which is locally in agreement with
quantum physics but assumes no global causal order. They have further shown
that there exist \emph{non-causal} correlations originating from
\emph{inseparable} process matrices that violate a \emph{causal inequality}
(CI) derived under the assumption that events are ordered with respect to some
global causal relation. This CI can be understood as a guessing game, where two
separate parties, say Alice and Bob, generate random bits (say input bit) in
their respective local laboratories. Bob generates another random bit (say
decision bit) which determines their goal: whether Alice has to guess Bob's bit
or vice-verse. Here we study this causal game but with biased bits and derive a
biased causal inequality (BCI). We then study the possibility of violation of
this BCI by inseparable process matrices. Interestingly, we show that there
exist \emph{inseparable} qubit process matrices that can be used to violate the
BCI for an arbitrary bias in the decision bit. In such scenario, we also derive
the maximal violation of the BCI under local operations involving traceless
binary observables. However, for biased input bits, we find that there is a
threshold bias beyond which no valid qubit process matrix can be used to
violate the causal inequality under \emph{measurement-repreparation} type
operation.
