A few key ideas about radioactivity
Most radioactive sources are made in nuclear reactors
Almost all the isotopes that are used in medicine and industry are made in nuclear reactors rather than being found naturally. These nuclear reactors are small and have normally been built specifically to provide useful radioactive isotopes.
In a nuclear reactor big nuclei like uranium are split into smaller ones that are always radioactive. These are separated out using chemistry to isolate the ones we want.
Gamma radiation only ever follows alpha or beta decay
An alpha or beta particle is released because the nucleus changes its composition.
This often leaves the nucleus with excess energy which it loses by emitting a gamma ray. This simply changes the way the protons and neutrons are arranged.
One nucleus decaying gives one particle
When a nucleus decays it changes to become more stable. As part of this change a particle, like an alpha or beta is emitted. If the nucleus after the decay is fully stable then there is a one-to-one relationship between the decay of a nucleus and a particle being emitted.
Every decay produces one and only one particle (if you ignore any gamma radiation). So if you see lots of particles being given off this means the nuclei must be becoming stable very quickly.
Long half-life means low radioactivity
Something is only radioactive while there are unstable nuclei around to decay.
If it's going to be radioactive for a long time then the nuclei can't be decaying very fast, which means particles can't be given off very often. This means it's not very radioactive.
Conversely, if something is very radioactive then all the nuclei quickly become stable and it stops being significantly radioactive in a short time.
Unstable nuclei don't grow old
An old person has a greater chance of dying this year than a young person but this isn't true of nuclei.
The chances of a nucleus decaying this second are exactly the same no matter how long the nucleus has already been around for. No matter how short the half-life is, all you can say is how likely a nucleus is to decay each second. And every second it's exactly the same.
The nucleus of an isotope with a half-life of a billionth of a second can still be around in a billion years and the nucleus of an isotope with a half-life of a billion years may only last a billionth of a second before decaying.
With large numbers of nuclei you can always say that more of the nuclei of an isotope with a short half-life will decay than nuclei of an isotope with a long half-life. You simply can't predict in advance which particular nuclei will do the decaying.
Cancer is completely random
Everyone gets radiation sickness above a certain dose but this isn't true of cancer. All you can say with cancer is that the higher your exposure, the greater the risk. Bigger exposure to radiation won't give you worse cancer.
Someone exposed to a tiny dose may get cancer but someone exposed to a much larger dose may escape it. All you can talk about is probability.