4 ms·
The asymmetry is that if you pick the one with the car first (only 1/3rd chance) then Monty Hall can show you either of the other doors. If you pick one withou
by lamontcg 3y ago
The asymmetry is that if you pick the one with the car first (only 1/3rd chance) then Monty Hall can show you either of the other doors. If you pick one without a car first (2/3rd chance) then Monty Hall must show you the door that doesn't have a car and the remaining door always has the car.
So 2/3rd of the time you have a 100% chance of a car if you switch, 1/3rd of the time you have 0% chance of a car if you switch. So switching is a 66.7% probability of a win and is the 'inverse' of if you picked the car right on the first guess or not.
The choice of which door Monty Hall shows you is constrained and isn't an independently random event since he never shows you the car when he opens a door.
You can also laboriously go through every event and "bifurcate the world" and then add up the probabilities, but you need to be careful about how you do it:
1. assign the car to a door (there are three possibilities)
2. have the contestant pick a door (there are now nine possibilities)
3. handle which door Monty Hall shows (this is tricky because for the three permutations in #2 where the contestant picked the car, then the door Monty Hall picks is unconstrained and so the world splits, but you are slicing a 11.1% chance in half for ~5.6%, while for the six permutations where the contestant didn't pick the car, Monty Hall is constrained so those remain 11.1% chances (at every level the probabilities have to add up to the probabilities before them and you can't violate 100% when you sum across all of them).
4. then have the contestant either switch or not switch and cut all your probabilities in half again.
5. now color everything according to winning or not and sum it all up.
I call that the "Quantum Many Worlds Interpretation Solution to the Monty Hall Problem" and if you do step #3 correctly you should get vos Savant's answer.
- Lendal 3y agoAha! This is the first explanation that reveals the most common error. The fact that the reveal isn't just a random door. The host must not reveal the car. That changes everything. I believe that's what most people are missing. But if you actually stated that explicitly, you would give away the answer and the problem wouldn't be fun anymore. The fun is in not explicitly stating every unconscious assumption you have about the problem.
- calderknight 3y agoThat's clear. Otherwise the question would just be about speculating on the host's personality.
- munificent 3y agoYou would be surprised how many people do correctly understand the problem (that the host must open a door and must not reveal the car) but still get the resulting probability wrong.
- calderknight 3y agoIt's simple. To begin with it's 1/3 your pick is right. You know that a goat will be revealed. So it's absurd to suppose that the fact that a goat is revealed has increased the probability of your initial guess being correct. Which door is opened to reveal a goat is totally irrelevant, as either would (under the absurd supposition) increase the probability of the initial guess to 50%. In other words, an increase to 50% would be guaranteed from the very beginning. So a contradiction is reached with the obvious fact that initial prob is 1/3.
- elteto 3y agoFantastic answer. I went through and built the tree and it just shows everything so neatly. There's just a small mistake, when you say: 4. then have the contestant either switch or not switch and cut all your probabilities in half again. You don't halve probabilities again because the strategy we are evaluating is _to switch_. So we always switch. It is deterministic. If we switch deterministically we end up with 6 cases (leaves in the tree) with probability 1/9 where we win. 6/9 = 2/3! If we switch randomly then we do split probabilities again, and end up with 6 cases with probability 1/18. 6/18 = 1/3.
- lamontcg 3y agoI don't really consider that a mistake it lets you compare the switching strategy to the not-switching strategy. YMMV.