Monty Hall Solution

If you haven't done so already, take a look at my previous post and try to work it out on your own then come back to this one. The intuitive answer would be that since there are now only 2 doors to chose from each has a 50/50 chance of winning, however that is actually wrong, the correct answer is sticking gives a 1/3 chance of winning but switching gives a 2/3 chance of winning, there are several different ways to explain this and I will have a go myself before giving up and linking elsewhere.

The shortest analysis would be to say that the door the player chooses has a 1/3 probability and the other two combined have 2/3, then when one door is opened revealing a goat the 1st door still has a probability 1/3, the open door changes to zero leaving the remaining door with a probability of 2/3.

A more developed analysis takes a closer look at all the possibilities, for the sake of this we will call the doors A, B and C, say the car is behind C (the contestant is unaware) there is a 1/3 chance of the contestant selecting each door initially, so the total probability of all options that start with each door choice is still 1/3. If the contestant chooses A first the host will open B or C, but C has the car so his only choice is to open B in this situation accounting for 1/3 of the total switching would win and sticking would lose. If the contestant chooses B first the host will open A or C, but C has the car so his only choice is to open A so again we have a situation covering 1/3 of total possibilities in which switching would win and sticking would lose. Now consider the contestant chooses C, the host can open either A or B at random, each has half of the initial 1/3 probability of outcomes where the contestant started with C, so both the contestant choosing C then the host opening A and the contestant choosing C then the host opening B have a 1/6 probability and will result in loss for a switch but a win for a stick, adding up all the probabilities we get 2/3 situations switching wins and 1/3 switching loses.

It may also help to imagine playing with say 100 doors, in this case we can all agree the first guess has a chance of 1/100, then the host opens all other doors except one revealing 98 goats and leaving two closed doors one with the car and one with a goat, hopefully the change in the scale helps illustrate how switching is beneficial.

Still not convinced, try it for yourself, get a friend and 3 playing cards from the same pack but with one that stands out, say a pair of reds and the ace of spades, the objective of the contestant is to pick the ace. You play the host and your friend plays the contestant, lay them face down get your friend to pick one, then turn over a red and ask if they want to stick or swap and compare several say 10+ trials where they swap to the same number where they stick, once they know the game you can swap places, maybe make it a competition one of you use one strategy and one use the other and see who comes off best. To make it a little more interesting make it into a drinking game, when the player wins the host drinks when the player loses they drink. Just remember 10 is a small sample so it is possible that you will not be able to tell the difference, if that is the case continue for a while and it will clear up.

This has a few pictures to help follow the argument.
Remember it is best if you truly understand what I'm getting at here so have a think about it, maybe come back in a day or so for another attempt. Admittedly is a hard concept and many people you would expect to be able to make sense of it struggle so admitting you don't quite get it is fine too just as long as your honest about it, it is certainly better than blindly rejecting my argument because you don't understand or even blindly accepting just because I look like I know what I'm talking about.