The 2 goats didn't make sense to me but the host perspective does I think, like: after player's first choice, there's two cases. Either player picked the car,

The 2 goats didn't make sense to me but the host perspective does I think, like: after player's first choice, there's two cases. Either player picked the car, or they didn't. If player picked the car, then host has two choices to pick from, if player didn't pick the car, the host has 1 choice to pick from. (At this point you can sort of imagine a binary tree with 3 leaves and 5 nodes) So either player picked the car the first time with a 1/3 chance (so, less likely), or didn't (more likely). So it's more likely for player to miss the first time (unless you're a precog psychic like me), which, "paradoxically (but not really)", by missing (by failing), player actually improves odds for second guess. So, on average the second guess is more likely to hit, because, on average, the first guess is more likely to miss. Now finally it makes sense to me. Because it's more likely to miss on first guess, it's also more likely to hit on second guess, so it's more profitable to switch, on average. Do I get it now? I think I do! Woot! :P :) xx