Since I got taught about the Poisson distribution I learned to modify $\lambda$ instead of $x$ as a "rule" while solving problems, however, I never was taught the logic behind it which is why I am posting here, to clarify what I am talking about here is an example:
A company has $100$ independent maintenance centers that received service calls. On average, the number of service calls received by each maintenance center is $6$ per hour. Find the probability that, for a given maintenance center, more than $30$ minutes will elapse before receiving two calls?
The way to solve this would be by converting the mean per hour to become mean per $30$ minutes (so $\lambda=3$) then use the Poisson distribution to find $P(X<2)$ which yields $\approx 0.199$
An incorrect way to do things would be saying that "More than $30$ minutes elapsing before receiving two calls means that more than $60$ minutes elapse before receiving $4$ calls" then use our default $\lambda=6$ to find $P(X<4)$ which yields $\approx 0.151$.
While I know that the first approach is correct and the second is incorrect I don't know why that is the case. Any help would be appreciated.
You are not "modifying $\lambda$", you are computing the expected number of events, which is $\lambda$. You are given a rate of $6$ per hour, but that could be expressed as $3$ per half hour or as $\frac 1{10}$ per minute. $\lambda$ does not have any units of time in it. As the question asks about things happening in $30$ minutes, you need the expected number of events in $30$ minutes.
Your incorrect approach would count times there were three events in the first hour but they all happened in the first $20$ minutes. That is not an event we want to count.