Yes you should switch due to the fact since the host knew that the door he open would have a goat thus boosting your probability of winning to 2/3 if you switch. why? well the are actually multiple ways to prove this but I find the most understandable explanation to comes from Bayes Theorem and the use of conditional probability. Essentially Bayes Theorem states that we can use the occurrence of a previous event to ascertain the probability of a subsequent event in a similar context. Here is how it applies to the Monty Hall Problem. when you first choose a door you have 3 options so you have a 1/3 of picking the door with the grand prize. Now the host will purposely pick one of the two doors with a goat behind it of which there are two options so he has a (1/2) chance of picking the door he did. he then asks you if you would like to change your selection to the other unopened door if you do so the probability of the car being behind that is 2/3. this is because we are making a now informed decision knowing that The host has given you information on a door he knew had a goat. You now know that in yore initial selection you had 1/3 chance in being correct and a 2/3 chance of being wrong but since we know that of of the doors we did not choose has a goat the entirety of the 2/3 probability now rests on the door we can switch to. below is a quick proof of the concept using Bayes theorem.
- P(H) is the prior probability that door 1 has a car behind it, without knowing about the door that Monty reveals. This is 1/3
- P( not H) is the probability that we did not pick the door with the car behind it. Since the door either has the car behind it or not, P(not H) = 1 - P(H) = 2/3
- P(E given H)is the probability that Monty shows a door with a goat behind it, given that there is a car behind door 1. Since Monty always shows a door with a goat, this is equal to 1
- P(E given not H) is the probability that Monty shows the goat, given that there is a goat behind door 1. Again, since Monty always shows a door with a goat, this is equal to 1