In a dark room, 3 blue hats and 2 green hats sit on a table. 3 people, 1 blind, enter the room. Each of the 3 people takes a hat random, without being able to see what colour they wear. The remaining 2 hats are taken out of the room.

The light is switched on and each person is asked to say whether they can guess the colour of the hat they are wearing.

– The first person says NO

– The second person, after the first one, says NO

– The blind person speaks last.

Which of the following sentences related to the blind man must be true?

1 – Probability his hat is blue is 60%

2 – He’s wearing a blue hat

3 – He’s wearing a green hat

4 – Probability his hat is green is 20%

Right answer is: 2) The blind man is wearing a blue hat

There are only seven possible combinations of hats using two green (G) and three blue (B) hats [notation is (FirstPerson,SecondPerson,BlindPerson)]:

(BGG) , (GBG) , (BBG) , (BGB) , (GGB) , (GBB) , (BBB)

Scenario 1 (BGG) : That is not possible. Indeed, since there are only two green hats, the first person would have seen the other two wear a green hat and would have known strainght away he was wearing a blue one.

Scenario 2 (GBG) : Similar to Scenario 1, thesecond person would have known he was wearing a blue hat. therefore this scenario is not possible.

Scenario 3 (BBG) : The second person knows that if he and the blind man had both been wearing green then the first one would have known he was wearing a blue hat and would therefore have answered “Yes”. Since the first person answered “No” and the blind man is wearing a green hat, then the second person would have known he was himself wearing a blue hat. Therefore this scenario is also not possible.

The remaining four scenarios are (BGB), (GGB), (GBB), (BBB), all of which have the blind man wearing a blue hat.

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