TITLE: A question about three forcings
QUESTION [8 upvotes]: Let $P_1$ be the finite support iteration of random forcings of length $\omega$. Let $P_2 = \text{Random} \times \text{Random}$ and $P_3 = \text{Random} \times \text{Cohen}$. Are $P_i, P_j$, for $1 \leq i < j \leq 3$, forcing isomorphic? All I know is that all of them add a $P_3$-generic.

REPLY [3 votes]: As you mentioned in a comment, in a $P_2$-extension there are two random reals the sum of which is Cohen. However, forcing with Cohen over $V[\mathrm{Random}]$ doesn´t add any real that is random over $V$. So the sum of two random reals in a $P_3$-extension cannot be Cohen, since Random doesn´t add Cohen reals.
This argument can be found in Perfect sets of random reals by Brendle and Judah, at the beginning of section 3.