Finding exact $p$ value for the sign test, using the table with probabilities under the binomial distribution
Assuming a table for a certain number of trials $n$, with a column per true probability $P$, and a row for each possible number of successes $X$
Two sided 
$p$ value is the probability of finding the observed number of positive differences or a more extreme number, given that the null hypothesis is true.

Right sided 
$p$ value is the probability of finding the observed number of positive differences or a larger number, given that the null hypothesis is true.

Left sided 
$p$ value is the probability of finding the observed number of positive differences or a smaller number, given that the null hypothesis is true.
