  We study the properties of algebraic independence and pointwise algebraic
independence in a class of continuous theories, the randomizations $T^R$ of
complete first order theories $T$. If algebraic and definable closure coincide
in $T$, then algebraic independence in $T^R$ satisfies extension and has local
character with the smallest possible bound, but has neither finite character
nor base monotonicity. For arbitrary $T$, pointwise algebraic independence in
$T^R$ satisfies extension for countable sets, has finite character, has local
character with the smallest possible bound, and satisfies base monotonicity if
and only if algebraic independence in $T$ does.
