How could I calculate the chance of a proton actually joining the nucleus and 2 protons and 3 and so on? Assuming a large number of protons are fired at a substance?
What you're looking for is the absorption cross section $\sigma$ of the target. This quantity has units of area, and is typically measured in barns (1 barn = 1 b = 10-28 m2.) There are many such cross-sections one can define, for any particle interaction you might care to name. If you have a beam of protons with a flux of $f$ (i.e., $f$ protons per area per time), and you send this beam at a target for a time $\Delta t$, then the expected number of absorptions is $$ \langle N \rangle = f \sigma \Delta t. $$
Calculating these cross sections from first principles is basically impossible (but then, that's true of most calculations in nuclear physics.) Instead, they must be experimentally measured, or looked up in the literature. Note that the cross section usually depends on the energy of the incoming protons; it also depends on the target, so once the target has absorbed one proton, the cross-section to absorb a second proton will be different since the target has changed.
Finally, note that proton capture is relatively rare compared to the more common neutron capture, since the proton and the target are both positively charged and tend to repel each other. Cross sections for scattering of protons off of nuclei can be defined in much the same way. There may not be a huge amount of data out there for proton capture cross sections, compared to either proton scattering or neutron capture.