Let's look at each strategy one at a time. Before we start, it is important to know about complementary events. Complementary events are when an event doesn't happen. For example, if the event is 'rolling a six on a dice,' the complementary event to that is 'not rolling a six on a dice.' The probability of an event plus the probability of it's complementary event always equals one, because the event either happens or it doesn't, which makes it certain. You can use this fact to find the probability of an event if you know the probability of it's complementary event, with the formula P(Event)= 1-P(Complementary Event). Complementary events are useful for all sorts of other reasons, too.
Your chance of picking the car before Monty opens a door is
P(car behind your door)=1 outcome/3 outcomes
P(car behind your door)=1/3
Once Monty opens a door, because you don't switch the chance of the car being behind your door is still 1/3. You might think it become 1/2 because there are now only 2 doors in play, but remember that just because there are two outcomes doesn't mean they have equal probabilities of happenining! Also, you first picked when there are three options, and because you are ignoring the new information, it doesn't change the probabilities.
Your chance of losing is P(lose)=1-P(car behing your door)
P(lose)=1-1/3
P=2/3
Your probability of picking the door with the car behind it first is still P(car behind first door)=1/3. Using complementary events, we can work out that
P(car isn't behind first door)= 1- 1/3
P(car isn't behind first door)= 2/3
That isn't very helpful in the first step of the game though, because there are two other doors, each of which the cars has a 1/3 chance of being behind. But then Monty opens a door and reveals a goat. Your chance of having guessed right is still 1/3, and your chance of having guessed wrong in still 2/3. But there's only two doors in play now. There was a 2/3 chance you were wrong, but now instead of that probability being spread between two equally likely doors, it's been 'concentrated' behind one. So there's a 2/3 chance the car is behind the other. If you switch, there's a 2/3 probability of you being right now. Hooray!
(C) Zoe Hills, 2016