In the Monty Hall problem the player can choose one of three doors. One hides a valuable reward, the others, goats. Once the player chooses a door, one of the other doors is revealed to have a goat behind it. The player is then given the chance to switch their choice to the other unopened door.
It is now accepted that switching does improve the players chances and this is shown to hold for any number of doors and any ratio of goats and rewards.
Does the above imply that whenever I’m choosing an item from a set of multiple items and I’ve made up my mind which item I want, but before I reach for it I’m shown one of the other items is undesirable I should change my mind and select some other item from the set?
For instance, me and my friend are applying for jobs. We both hold offers from the same 10 companies and we both like all of the offers equally. We both decide which company each of us wants to work for arbitrarily (we choose different companies). It is assumed that some of the positions will be good, while others will turn out to have bad work life balance or toxic environment. We don’t know the exact odds as we don’t have a perfect knowledge of the market. My friend is a little ahead of me in the process and starts work a month before me in company #6. Just before I sign my own contract with company #4 I get a call from my friend about terrible work culture at company #6. I have seemingly received no new information about company #4, but by the Monty Hall principle I should still reroll my choice and accept a different offer from the 10 I received but not from company #4 or #6.
Is my reasoning above correct? If not, why? The situation above seems analogous to the Monty Hall problem as:
- I need to choose an item from a set of multiple items
- I know some of the items are good and some are bad (I don’t know the exact odds, but does that matter? Would anything change if another friend, that has a glassdoor account, told me that there are 5 good companies and 5 bad ones?)
- I choose one of the items (company #4) and then I receive information about one of the bad items (company #6) and am given one last chance to change my choice
I think there is a misunderstanding in why you should change doors. The reason why you should change doors is that the hosts knows where the car is. That is, the host will certainly not open the door with the car behind. This means new information becomes available once the host opens the door. Heuristically, this information is utilized by changing the choice.
Hence the answer to your question "Does the above imply that whenever I’m choosing an item from a set of multiple items and I’ve made up my mind which item I want, but before I reach for it I’m shown one of the other items is undesirable I should change my mind and select some other item from the set?" is no.