Q: Adam, Brian, Cathy, Doug and Eve are at a round table for five when they realize that they are in age order around the table. What are the odds of that?”

A: The odds are 11:1 (against) and the probability is 1/12.

If we imagine that they sat down in age order randomly:
The first person is random.

The second person can sit to the first person’s right or left so there are two slots and 4 people left. (2/4)
The third oldest now has 3 chairs to choose from = 1/3
This leaves two seats for the 4th person to choose from (1/2)
The 5th person has no choice as there is only 1 seat left.
Therefore the probability is:
(1) (2/4)(1/3)(1/2)(1) = 1/12