We have known many proper (non-ccc) forcing notions which do not add random reals. We note that a sigma-centered forcing notion is a ccc forcing which does not add random reals.
We introduce two properties of forcing notions, and show that a forcing notion with one of these properties does not add random reals.
This is an extension of Zapletal's paper: Keeping additivity of the null ideal small. Proc. Amer. Math. Soc. 125 (1997), no. 8, 2443-2451.