As a test to see if his advisors are worth their pay, the king devises the following test. Each of his advisors is to be sequestered incommunicado in a separate room. One by one, in random order, they are to be summoned to the throne room and then sent back to their chambers. Upon arriving in the throne room, any advisor may choose to end the test. At that point, if every advisor has been called in at least once, they pass. Otherwise, they fail and $bad_consequence.
The throne room contains a statue, which can rest either right-side-up or upside-down. It is initially right-side-up, and advisors are permitted to flip it while they are in the throne room. The advisors are allowed to strategize freely before the test begins, but once it begins, this statue is their only means of communication.
The advisors' probability of being called at a given time are not necessarily equal. The only thing specified about the order in which they are called is that any given advisor will eventually be called any arbitrary number of times given that the test goes on long enough (that is, the limit as time approaches infinity of the number of times that an advisor is called is infinity). Also, the timing is random, so advisors aren't able to deduce their position in the sequence.
What strategy can they use that will assure success?