GATE Electronics and Communications (EC) 2015 Shift 1 Solved Paper

A source emits bit 0 with probability 1/3 and bit 1 with probability 2/3. The emitted bits are communicated to the receiver. The receiver decides for either 0 or 1 based on the received value R. It is given that the conditional density functions of R  are as
${\mathit{f}}_{\left(\mathit{R}∕0\right)}\left(\mathit{x}\right)=\{\frac{1}{4},-3\le \mathit{x}\le 1.\mathit{a}\mathit{n}\mathit{d}{\mathit{f}}_{\left(\mathit{R}∕1\right)}\left(\mathit{x}\right)=\{\frac{1}{6},-1\le \mathit{x}\le 5.$
The minimum decision error probability is