4 ms·
> After the host's hidden knowledge has reduced the problem space from N options to 2 options, the probability of getting the car is increased from 1/N to 1/2.
by henrikschroder 7y ago
> After the host's hidden knowledge has reduced the problem space from N options to 2 options, the probability of getting the car is increased from 1/N to 1/2.
No, your numbers are wrong.
In your initial choice, the probability of picking the winning door is 1/N.
When Monty opens doors so that there is a single door remaining, the probability of that door containing the prize is the complement of the probability that you picked the winning door, i.e. 1-(1/N).
So switching doors increases your probability from 1/3 to 2/3 if N=3, not to 1/2.
If N=1000, switching doors increases your probability from 1/1000 to 999/1000, and that is why increasing the number of doors makes the correct solution more intuitive for some people.
- CydeWeys 7y agoAnd the point I'm making is that it's not clear at all why opening 998 doors out of 1,000 is analogous to opening 1 door out of 3 -- why wouldn't you still be opening just the 1 door? If you pick one of 1,000 doors, and then the host reveals a goat behind a single door, there's a slight (but only very slight) advantage to switching. The advantage is most pronounced with the fewest possible number of doors, i.e. 3. That's why this approach to teaching it has never made sense to me -- it's not clear at all why opening 998 doors should be analogous to opening 1 door, which it's much more obvious that opening 1 door is analogous to 1 door.