Response to Monty Hall Problem
Statistical information is based on the probability of something to happen. Even though it does not depict certainty but just the odds of something to happen, it still holds value in my opinion. If the statistics say odds are 50% or even 75% then I might not get influenced by them, but when it says there is 80% or 90% probability like in the case of disease example and the users of Acme toothpaste then I would consider it is a very healthy trend. It is not advantageous to switch to door 2 after having selected door 2 at the first place. The reason why I will be convinced is that the percentage population having disease occurrence and toothpaste usage is substantial. The reason is that the probability of the prize car being behind door 1 before the host Monty Hall opened door 3 was 1/3 or 33%. After having opened door 3, the probability of prize car being behind door 1 grew to become ½ or 50%. The player can switch to door 2 now, or stay with the door 1 without this decision making any difference on the odds of him winning the prize car. There is an assumption that the host who knows behind which door is the car at does not change the location of the prize car after opening the gate 3. Another assumption is that the odds for the prize car being behind any of the doors 1, 2 or 3 are even.
Response to Student Post
There is no confirmation if what the statistics say will inevitably be so. However, statistics does tell about how much chance does there exist of something to happen. As part of human tendency, when there is a high percentage of something or, in other words, a high probability, a person would avoid if it depicts risk and will take it up if it depicts benefit. In neutral matters like usage of toothpaste, if almost every person is using the same toothpaste then again the human tendency kicks in. There must be something about this toothpaste making it so popular amongst people. Since the odds of prize car being behind either door 1 or door 2 are even at 50%, hence the player will not find any use of switching to door 2 changing the previously selected door 1.