Is there such a thing as chance?

No, not really.We carbon-date things (and do other kinds of radiometric dating) by measuring how incredibly large numbers of unstable nuclei act. But we do not carbon-date things by looking at one particular unstable nucleus.If you have one atom of carbon-14, you cannot predict when it will decay. You can calculate a probability that it will decay in some specified time period, but for that one nucleus that's the best you can do. It may or may not actually decay in that time period.However, when you have trillions of carbon-14 atoms, or (as is more usual) trillions of trillions of trillions of carbon-14 atoms, you can predict extremely precisely how many of those nuclei will decay in any specified time period ... but you cannot predict which of those nuclei will be the ones that decay. You know that some of them will decay but not the particular ones that will decay.Radioisotope dating methods work because any samples we take, even the ones that are incredibly tiny by our everyday standards, have enough atoms in them that the large-number-of-atoms statistics work very precisely.Or, as it says at The Law of Radioactive Decay:"You can't, however, predict the time at which a given atomic nucleus will decay. For example, even if the probability of a decay {of one particular nucleus - JRF} within the next second is 99 %, it is nevertheless possible (but improbable) that the nucleus decays after millions of years."That site has a nice Java applet that simulates decay.

I don't like the word "assumption" in this context because it implies "taken for granted or accepted as true without proof", which is not so. I prefer "premise", because the premises involved in radioisotope dating have been checked six ways from Sunday. So carbon dating is based on premises. A slight overstatement, depending on what you mean by "accurate". It's good to about 1% for samples 5,000 years old and less, and the error rises slowly (and not smoothly, for complex technical reasons) until about 50,000 years old, at which point the level of C-14 is so low that our best instruments can barely distinguish it from the backgound noise. Other methods, based on other radioactive isotopes, are used for older samples.