The problem certainly becomes clearer if you state that Monty never opens the door with the prize. But I have to admit that even when presented with the Monty

The problem certainly becomes clearer if you state that Monty never opens the door with the prize. But I have to admit that even when presented with the Monty Hall problem extension, 100 doors, with Monty offering 99 chances to change, it was still not "intuitive" to me that one should change. I don't think it is just poor wording. And the idea that probabilities can shift based on Monty's knowing where the prize is certainly not intuitive, in part because many of us do not intuit accurately about probability.