The chance the cat could die or stay alive were actually close IIRC since only a tiny radioactive substance was present and there was a solid chance that it would not have killed the cat, basically in the case none of its atoms would of decayed, while instead even a single one decaying could have killed it by triggering the hammer breaking the poison container.
The paradox though is that in effect a cat could never be alive and dead at the same time (quantum superposition of two eigenstates is the specific term) in the same universe and it seems that quantum mechanics presents problems when applied to the macroscopic world. It basically highlights the problems emerged with the Copenhagen interpretation which is funnily still the most popularly accepted amongst physicists. They think of decoherence, so basically that is why the macroscopic world doesn't act as counterintuitively as the quantum one.
A theorical solution to this problem is provided by Hugh Everett's Many Worlds Interpretation of quantum mechanics by which the two cats simply branch into different worlds, parallel universes basically, leaving just one cat per world. Other than the paradox of the cat it also addresses the measurement problem by claiming that all the outcomes of the wave function are correct and there is no collapse of the wave function. Physicists still nowadays struggle to understand whether or not there is a collapse of the wave function (basically the superposition of the eigenstates I mentioned, collapsing into a single one) and they don't know if and what makes it collapse exactly. Most physicists believe that MWI doesn't fundamentally put an end to all the problems with quantum mechanics, but since it at least addresses the paradox Schrödinger highlighted in a good way it's turning more and more popular after many theoretical physicists are showing why the likelihood of a multiverse is there and we don't just live in one single bubble.
There's a huge debate on this stuff still going on, I became passionate long ago and nerded the hell out of it.