The Monty Hall Problem

The Monty Hall Problem is a brain teaser based around the American TV game show Let's Make a Deal and named after that show's original host. The general premise of the problem is as follows:

You are presented with three doors. Behind two of the doors, there is a goat. Behind the third is a prize. You are asked to make a guess at which of the three doors contains the prize. Monty Hall then opens one of the doors to reveal a goat. You are given the option to keep your original door or switch doors.

Which one is more likely to result in you winning? Are the chances equal?