GATE DA 2025 Q41: Floor of an exponential random variable is geometric

For $x \in \mathbb{R}$, the floor function is denoted by $f(x) = \lfloor x \rfloor$ and defined as follows

$$\lfloor x \rfloor = k, \quad k \le x < k + 1,$$

where $k$ is an integer. Let $Y = \lfloor X \rfloor$, where $X$ is an exponentially distributed random variable with mean $\dfrac{1}{\ln 10}$, where $\ln$ denotes natural logarithm. For any positive integer $\ell$, one can write the probability of the event $Y = \ell$ as follows

$$P(Y = \ell) = q^{\ell}(1 - q)$$

The value of $q$ is