What are the chances of false negative mris and cat scans? what are the odds that an mri and a cat scan would both show no cancer when there really is cancer present? what about the opposite, where they would both should cancer when there really is none?

This : This depends widely on what part of the body is being imaged, what type of cancer is present (or hopefully not present), how large the cancer is, and what type of MRI or ct is performed. If the proper imaging study is performed to evaluate for a specifically suspected cancer then the percentage of cancers that are discovered is very high and the "false" cancers detected is quite low. The best imaging study must be determined on a case-by-case basis by yourself, your primary care provider, and often a radiologist. I strongly advise against having any imaging study performed if you have no symptoms or only vague, nonspecific symptoms. Not only does this often lead to unnecessary cost, but also greatly increases the chances of falsely finding a "cancer". This in turn can lead to additional expensive testing and a lot of unnecessary emotional distress.

No test is 100%: The sensitivity of MRI and ct scans depends partly on the type of cancer and the location in the body. In general, they can find tumors that are a few millimeters in size which can be 10's of millions of cancer cells. If all scans are negative a patient may be considered in remission, but with cancer it is really only the "test of time" that ensures no microscopic cells are left.