In random sequential adsorption (RSA), objects are deposited randomly, irreversibly, and sequentially; attempts leading to an overlap with previously deposited objects are discarded. The process continues until the system reaches a jammed state when no further additions are possible. We analyze a class of lattice RSA models in which landing on an empty site in a segment is allowed when at least $b$ neighboring sites on the left and the right are unoccupied. For the minimal model ($b=1$), we compute the full counting statistics of the occupation number. We reduce the determination of the full counting statistics to a Riccati equation that appears analytically solvable only when $b=1$. We develop a perturbation procedure which, in principle, allows one to determine cumulants consecutively, and we compute the variance of the occupation number for all $b$.