Say I have a Poisson distrribution with mean $7$ per hour. And I want to calculate the probability of a certain number say $2$ happening in that hour. Then the probability would be $\frac{e^{-7}7^2}{2!}$.
Now if I want to calculate the probability of $2$ in half an hour- I could decide to say the mean is really $3.5$ per half hour and then do $\frac{e^{-3.5}3.5^2}{2!}$.
Or I could keep the mean at $7$ per hour and instead say $2$ per half hour is really just the same as $4$ per hour so do $\frac{e^{-7}7^4}{4!}$. As can be guessed these do not give the same answer- when there's an exponential and a factorial term involved- no linear factors would cancel to give the same result as before.
So which is right? Remember I want to calculate the probability of $2$ in half an hour. Thanks.
The number of events in a half-hour period has Poisson distribution mean $\frac{7}{2}$. So your first calculation is correct.