The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. Thus, Poonen's Bertini theorem over finite fields has a motivic analog due to Vakil and Wood, which expresses the motivic density of smooth hypersurface sections as the degree goes to infinity in terms of a special value of Kapranov's zeta function. I will report on joint work with Sean Howe, providing a broad generalization of Vakil and Wood's result, which implies in particular a motivic analog of Poonen's Bertini theorem with Taylor conditions, as well as motivic analogs of many generalizations and variants of Poonen's theorem. A key ingredient for this is a notion of motivic Euler product which allows us to write down candidate motivic probabilities.