How reliable are intuitive answers?

Here are two examples I found in a book I’m currently reading.

1) Try to think of an intuitive answer without spending much time:

A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?

2) Again, try to think of an intuitive answer without spending much time. Does the conclusion follow the premises, yes or no?

All roses are flowers.
Some flowers fade quickly.
Therefore some roses fade quickly.

(Source: D. Kahneman)

Is this why Lehman Brothers went bankrupt? Were their people relying on intuitive answers?

What is your own personal experience with intuitive answers?