Goat or Car Problem Confusion

I have spent a long while thinking about the goat or car problem, and I am totally perplexed.  I have gone over the tree diagrams over and over, and I just do not understand how switching your door choice to the other door would do anything for your chances at all.  Of course, we know, because we are looking at what is behind each door, which door should be chosen, but the person choosing the door doesn't know that, so I have no idea why your chances would be increased.  I've spent a lot of time thinking about this.

I don't see any logic involved in the tree diagram.  I am probably just confused, but this problem is totally and completely perplexing to me.  I decided just to write about my confusion because it was bugging me so much.  I would love for this to be better explained to me though.  I spent about 45 minutes looking at the tree diagram and trying to figure it out.  Everytime I thought I understood why the chance of winning a car would increase to 2/3 rather than 1/3 when the game show contestant changed their choice, I realized I was still just confused!

I would say that this was a successful blog prompt because it made me think. It challenged me and frustrated me, and challenging is what math should be.  I will continue to try and understand this problem.

The Probability of Missing a Gate

I thought about trying to solve the swimming pool problem, but I also wanted to relate probability to my daily (or almost daily) life.  I started thinking about the nerves that build up before a ski race.  You have to finish the race, and make it around every gate, so all of the racers ask themselves this question: What if I miss a gate?
I then started thinking, what is the probability that I will actually miss a gate in a given race?

One particular course that I have raced on has 40 total gates, alternating red and blue.  On any given gate, there are two options: I can either make the gate, or miss it.  So, the probability of me missing a gate is (1/2)(to the 40th power).  (*I don't know how to do the formatting on here).  So really, I shouldn't be that worried about missing a gate.

It is interesting to me how skewed our perception of probability is.  We think we know, and we worry about things, but the numbers show us that the reality really isn't that scary!