The probability distribution for the number of goals scored per match by Team A is believed to follow $X \sim Poi(0.8)$.
Independently, the number of goals scored by Team B is believed to follow $Y \sim Poi(1.6)$.
What is the probability that the total number of goals is greater than 2.4?
If I let $Z = X + Y$, would it be correct to state that $Z \sim Poi(2.4)$?
Then, $\mathbb{P}(Z = z) = \frac{\lambda^z}{z!}e^{-\lambda} = \frac{2.4^z}{z!}e^{-2.4}$. However, this doesn't seem to work for decimal numbers. What would be a better approach be? I am thinking of perhaps using a normal approximation with $\mathcal{N}(2.4, 2.4)$, which trivially makes the probability 0.5. However, this doesn't seem to work. What would be the best way to solve this? Thanks!
Your first step, i.e. to state that $Z \sim \operatorname{Poi}(2.4)$ is correct. You are asked to calculate $P(Z>2.4)$ (and not equal, as already highlighted in the comments). Since $Z$ takes only integer values (as you correctly observed) you can calculate the required probability as follows: $$\begin{align*}P(Z>2.4)&=1-P(Z\le 2.4)\\&=1-P(Z\le 2)\\&=1-\left[P(Z=0)+P(Z=1)+P(Z=2)\right]\\&=1-\left[e^{-2.4}\frac{2.4^0}{0!}+e^{-2.4}\frac{2.4^1}{1!}+e^{-2.4}\frac{2.4^2}{2!}\right]\\&=1-e^{-2.4}\left[1+2.4+\frac{2.4^2}{2}\right]=1-0.5697\\&=0.4303\end{align*}$$