difference between events represented by random variables X and X-1

Question

let X be a geometric random variable, representing the number of trails up until first heads occurs.
so, $X=4$ will represent the event $T_1T_2T_3H_4$
part a
given this information, why would $X-1=3$ represent the event $T_2T_3H_4$?
isn't $X-1=3$ same as $X=4$? so, by that logic shouldn't they represent the same event?
what event would $aX-b$ represent?
part b
assuming the parameter for X is p, how can $P(X=k,X>1)$ be represented in terms of p and k.