If $\frac{1+4p}{4},\frac{1-p}{4}$ and $\frac{1-2p}{2}$ are the probabilities of three mutually exclusive events, then 

We have $0\le \frac{1+4p}{4},\frac{1-p}{4},\frac{1-2p}{2}\le 1$

and $0\le \frac{1+4p}{4}+\frac{1-p}{4}+\frac{1-2p}{2}\le 1$

$\Rightarrow -\frac{1}{4}\le p\le \frac{3}{4},-3\le p\le 1,-\frac{1}{2}\le p\le \frac{1}{2}$

and $0\le p\le 4$

$\Rightarrow max\left\{-\frac{1}{4},-3,-\frac{1}{2},0\right\}\le p\le min\left\{\frac{3}{4},1,\frac{1}{2},4\right\}$

$\Rightarrow 0\le p\le 1/2$