A specialized game show has three doors. Behind one is a car, behind two are goats. You pick a door. Before revealing, the host (who knows where the car is) opens one of the other doors to reveal a goat. You are given the option to switch. However, the host is 'lazy' and only opens a goat door 50% of the time when he can; otherwise, he ends the game prematurely. Given that he opened a door and showed a goat, what is the probability that switching wins the car?