Geometric or Poisson

Question

Drought length is referred to as the number of consecutive time intervals in which the water supply remains below a critical value. Consider the drought length as a random variable, denoted as $Y$, which is assumed to have a geometric distribution with $p = 0.409$

1. What is the probability that a drought lasts exactly $3$ intervals? $(0.0844)$
2. What is the probability that a drought lasts at most 3 intervals? $(0.878)$

My try:

1. $(0.409)^3 = 0.06841$.

2. $(0.409)^3 + (0.409)^2 +(0.409)^1 +(0.409)^0$.

Answer 1

The geometric distribution is defined as $$P(Y=k)=p(1-p)^{k-1},\quad k\geq 1$$ Equivalently: $$P(Y>k)=(1-p)^k$$

The first question just asks $P(Y=3)$, and the second question asks $P(Y\leq 3)$, which we can write $1-P(Y>3)$.