How do you know if a product is organic?

Assuming that it's bought from a retailer rather than grown yourself then it will most likely say so. Most organic foods sell at a higher price or are seen as "better" by some people, so whoever is selling organic products are almost certain to make sure people know.

I guess that sometimes unscrupulous people claim something is organic when it's not and, of course, mistakes are made. If so, then there is no way to tell just by looking at the product. I think it will be rare for this to happen though.

you don't. even those labeled "organic" often are not completely so.
its very difficult to grow mass quantities of organic anything without getting contamination from farms near by due to wind drift, water run off, etc.

It is radio active. That how the US Government checks that alcohol is drinkable.
I would read the label or maybe look at the grower's licence to see if it was Certified Organic.