  We study approximation in the unit interval by rational numbers whose
numerators are selected randomly with certain probabilities. Previous work
showed that an analogue of Khintchine's Theorem holds in a similar random model
and raised the question of when the monotonicity assumption can be removed.
Informally speaking, we show that if the probabilities in our model decay
sufficiently fast as the denominator increases, then a Khintchine-like
statement holds without a monotonicity assumption. Although our rate of decay
of probabilities is unlikely to be optimal, it is known that such a result
would not hold if the probabilities did not decay at all.
