Amoeba forcing is defined as follows:
$\begin{equation*} (\mathbb{A}, \leq) := (\{X \subset 2^{\omega} : X \text{ is open and } \mu(X) < 1/2\}, \supseteq) \end{equation*}$.
It is claimed that this forcing notion adds a measure 1 set of random reals. I get that if $G$ is $(\mathbf{V}, \mathbb{A})$-generic then $2^{\omega} - \bigcup G$ is a perfect set of random reals over $\mathbf{V}$ of measure 1/2. But how do we recover a set of random reals of measure 1 from the generic filter?
Once you have a set $S$ of random reals with measure $1/2$, I think the Lebesgue density theorem lets you improve the measure as follows. (I'll work in $[0,1]$, but everything transfers to $2^\omega$.) For any positive integer $n$, there's a rational interval $I$ whose intersection with $S$ has at least $1-\frac1n$ of the measure of $I$. Linearly stretch $I$ to $[0,1]$ and you have a set $S_n$ of random reals with measure at least $1-\frac1n$. Now take $\bigcup_nS_n$.
(Actually, this is even easier in $2^\omega$: Take the set of all tails of sequences from $S$. As far as I can see, you still use the Lebesgue density theorem to see that this works.)