Physicians at a clinic gave what they thought were drugs to $840$ asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, $55$ % of the patients reported an improved condition. Assume that if the placebo is ineffective, the probability of a patients condition improving is $0.5$. For the hypotheses that the proportion of improving is $0.5$ against that it is $>0.5$, find the p-value.

$p =$