**UCAT Question Analysis - Decision Making Q38**

There is a dark room where three blue hats and two green hats sit on a table. Three people enter the room where the last person enterring the room is blind. Each of the three people takes a hat at random without being able to see what colour it is and wears it on their own heads. The two hats which have not been picked are taken out of the room.

Lights on, and they are asked to say whether they can guess the colour of the hat they are wearing.

The first person says NO, the second person says NO and the blind person is the last to speak.

**Which of the following MUST be true?**

A. The probability that the blind man's hat is blue is 60%.

B. The blind man is wearing a blue hat.

C. The blind man is wearing a green hat.

D. The probability that the blind man's hat is green is 20%.

**Answer and Explanation**

**B: the blind man is wearing a blue hat**

There are only seven possible combinatiohns of hat using two green (G) and three blue (B) hats, as follows:

Scenario 1: BGG. Impossible. Since there are only two green hats, should the first person have seen the other two wearing green hats, he would have know that he is wearing a blue hat.

Scenario 2: GBG. Similar to Scenario 1, the second person would have known what hat he is wearing should the other two are wearing green hats.

Scenario 3: BBG. The second person knows that if he and the blind man had both been wearing green then the first person would have known he was wearing a blue hat and would have answered YES. As 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. So this scenario is also not possible.

The remaining four scenarios are BGB, GGB, GBB and BBB. All scenarios have the blind man wearing a blue hat.

Drafted by Quincy (UCAT Prep)