There are two related but very different types of random variable that could be associated with this problem. We could let $T$ be the time between consecutive arrivals. Then it is (sort of) reasonable, and certainly common, to assume that $T$ has exponential distribution.
In general, $\Pr(T \le t)=1-e^{-\lambda t}$, where $\lambda$ is a parameter that measures the arrival rate. If we measure time in minutes, then in our case $\lambda=10$.
Equivalently, $Pr(T \gt t)=e^{-\lambda t}$. So the probability that there are no arrivals in a $1$ minute interval is $\Pr(T \gt 1)$, which is $e^{-10}$.
Similarly, the probability of no arrivals in a $15$ second ($1/4$ minute) interval is $e^{-10/4}$.
The probability of at least one arrival in a specific $15$ second interval is $1$ minus the probability of no arrivals. this is $1-e^{-10/4}$.
Or let $N$ be the number of arrivals in a time interval of fixed length. It is (sort of) reasonable, and certainly standard, to assume that $X$ has Poisson distribution.
Let's go with $N$, the number of arrivals in a $1$ minute interval. Then $N$ is said to have Poisson distribution with parameter $\lambda$, where $\lambda$ is the arrival rate. In general we have $\Pr(N=n)=e^{-\lambda}\frac{\lambda^n}{n!}.$
We can use the Poisson distribution to solve our problems. For example, if we are measuring time in $1$ minute units, the arrival rate is $10$, and $\Pr(N=0)=e^{-10}\frac{10^0}{0!}$. This is simply $e^{-10}$.
If we measure time in $15$ second units, the arrival rate $\lambda$ is $10/4$, and we can calculate as before.
The Poisson is more complicated but more versatile than the exponential, for it lets us find quickly, for example, the probability of $7$ arrivals in a $1$ minute interval.
Remark: The exponential random variable $T$ can take on, in principle, any positive real number as a value. It is said to have continuous distribution.
By contrast, the Poisson random variable $N$ only takes on non-negative integer values. It is an example of a discrete distribution.
It is fairly likely that at this stage you have been exposed to the exponential and not to the Poisson.