Analysis - Measure the positions of the balls, calculate the angles, etc.
Guess - Hit the ball as hard as you can in a random direction.
Intuition - Ability to pot a ball, with a reasonable degree of certainty, without getting the ruler and the calculator out.
It requires a bit of practice too. But then the same can be said with intuition, it needs experience.
Difference between intuition and guessing is simple. Guessing is when you have no information. Intuition is when you have 0<N<1 bits of information available (i.e. fractional information). And certainty is when you have M>= 1 bits of information.