Could someone explain intuitively to me why for a geometric distributed Random Variable $X$
$$P(X \ge k) = (1-p)^{k-1}$$
and
$$P(X \le k) = 1- (1-p)^{k}$$
I understand the pmf of the geometric distribution but don't completely understand why these formulas are true.
$P(X \ge k)$ means we have at least $k$ trials before the first success, which happens iff the first $k-1$ (independent) trials fail, so with chance $(1-p)^{k-1}$.
And $P(X \le k) = 1 - P(X\ge k+1)$ (complementary chances obviously: either at most $k$ trials or at least $k+1$, and not both) and we apply the previous one to get $1-(1-p)^k$.
