Question about Monty Hall if you already knew a bad door beforehand

If you pick the car first, as you say, the host has 50% chance of confirming you have the car, and 50% of giving you no new information.

If you pick the unknown goat first, the host gives you no new information.

Ultimately the host has a $\frac{3}{4}$ chance of revealing the goat you already knew about.

If you're picking between those two randomly, then in $\frac{1}{4}$ of all cases, you pick the car, then the host reveals the second goat, so you win the car. In the other $\frac{3}{4}$ of cases, you're back to the $\frac{2}{3}$ - $\frac{1}{3}$ situation, since in $\frac{2}{3}$ of those $\frac{3}{4}$, you had the goat to begin with, and in the other $\frac{1}{3}$ you had the car.

$$P(\mathrm{car}|\mathrm{unknown\,goat\,revealed}) = 1$$

$$P(\mathrm{car}|\mathrm{known\,goat\,revealed}) = \frac{P(\mathrm{car}\cap\mathrm{known\,goat\,revealed})}{P(\mathrm{known\,goat\,revealed})} = \frac{1/4}{3/4} = \frac{1}{3}$$

You can expect to get the car $\frac{3}{4}$ of the time by switching if the goat you knew about is revealed and staying if not.

Of course, if you pick randomly between the two unpicked doors, you're a fool, because picking the door you know to be a goat forces Monty to show you the other goat and hence brand new car :)

As a side note, wouldn't it be better to pick the door you knew had a goat? Thus Monty will have to reveal the other goat?

Your chance still improves even if Monty picks the same goat you already saw. That may seem odd, so I'll try to explain it intuitively.

Forget Monty Hall for a moment and think about another situation. Your friend has two coins. One's a regular, unbiased coin, the other is double headed. He tells you to pick one at random. Your chance of having picked the double-headed coin is now 50/50.

Now he flips the coin you picked 10 times, and every time it comes up heads. It's still possible you picked either coin, but now it's clearly more likely you picked the double-headed coin.

In fact, even if he just flipped it once and it came up heads, you're still a bit more likely to have picked the double-headed coin. If we label the coins "S" (single-headed) and "D" (double-headed), the combinations are:

SH
ST
DH1
DH2

So eliminating ST, there's now a 2/3rds chance that you have the double-headed coin.

Going back to Monty Hall, we can map this situation directly. Your choice between the two coins maps to the choice between the two remaining doors, and Monty's choice of which door to reveal maps to the coin flip.

If you picked the goat door, Monty has only one option- picking the door you already saw. This corresponds to picking the double-headed coin, where the flip only has the option to come up heads. If you picked the car door, Monty has two options, just like if you picked the fair coin, it could come up either heads or tails.

So the situation where Monty picks the door you already saw is just like getting heads on the first flip- it means there's a 2/3rds chance you have the goat door. So, just like regular Monty Hall, you should switch.