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.



A question about three forcings
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](http://arxiv.org/pdf/math/9209205.pdf) by Brendle and Judah, at the beginning of section 3.


