His promotion is on the cards.

The idiom "on the cards" means something is likely to happen or is being considered.

* Option A: due - While something that is "due" is expected, "on the cards" implies more of a possibility or likelihood, not necessarily a certainty.
* Option B: evident - "Evident" means obvious or clear. "On the cards" doesn't mean it's already obvious, just that it's being considered or is likely.
* Option C: certain - "Certain" means sure to happen. "On the cards" is not as strong as "certain"; it's less definite.
* Option D: probable - "Probable" means likely to happen, which is the closest meaning to "on the cards."

Therefore, "His promotion is on the cards" means his promotion is likely or probable.