IT-CESENATICO 1992 Problem 4

A jury of $9$ persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability $1/2$ for each of the two possibilities, and noone is allowed to be abstinent. Find the probability that a fixed juror will be a part of the majority. In the case of a jury of $n$ persons, find the values of n for which the probability of being a part of the majority is greater than, equal to, and smaller than $1/2$ , respectively. (For $n = 2k$ , $k +1$ votes are needed for a majority.)