# Blue Forehead Room Solution

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## Description

Do not watch before Blue Forehead Room. THIS IS THE SOLUTION!

More free lessons at: http://www.khanacademy.org/video?v=-xYkTJFbuM0

More free lessons at: http://www.khanacademy.org/video?v=-xYkTJFbuM0

My instinctive reaction the first time I heard the given solution was that 100 perfect logicians simply would not need to wait for 100 (or 1 million!) lights to go on/off before the Bs know they are B and leave. Think about it what logic would each be going through during all that time???? Note that Sal does not really explain how the logic actually works for such a large group, just that it always takes the same number of lights on/off as there are people in the group before the Bs know they are B and leave.

What if we look at this using the probability that any one in the room is, or is not, NB instead? After all, in a room of 1 million, when they look around and see either all Bs, or maybe one NB, each knows there is only 1 chance in a million that they, themselves, are NB.

Each logician would reach the following conclusions before the 1st LO:

1 Everyone I see is B, so I know that I am the only possible NB in the group (1NB-me).

2 If I actually am the only NB, the others will see me as NB and think they might be a 2nd NB.

After the 1st LO, no one has left, but each logician who sees everyone else as B (ie. s/he is the only possible NB) thinks:

1 - I know I am the only possible NB, and the 1st LO did not help me know if I am, or am not NB, so I must wait for one more LO to see what the others do.

2 If I am NB, the Bs I see now know that they must be B and will leave at the 2nd LO:

Each stayed behind for the 1st LO, thinking s/he might be a 2nd NB. S/he sees everyone else as B, and knows that no one else can be a 2nd NB. The fact that everyone stayed at the 1st LO meant that everyone thought s/he also might be the 2nd NB, which is impossible. Everyone who sees an NB (me) now knows s/he is B and will leave on the 2nd LO.

After the 2nd LO, each logician who sees everyone else as B can now conclude:

1 If everyone else left, that means I am the only NB.

2 If no one left, that means the others did not see me as NB, or they would have left at the 2nd LO.

3 Since no one left, no one is NB. We are all B and will leave at the 3rd LO.

Since the probability of any one logician being NB is not affected by the size of the group, the same logic applies to any size group.

Bottom line - if there are any NBs in the group, all the Bs leave on the 2nd LO and the NBs are left behind. If everyone is B, they all leave on the 3rd LO every time.