Question

In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it . Upon asking , the person replies the following :
"The result of the toss is head if and only if I am telling the truth".
Which of the following options is correct?

• A. The result if head
• B. The result is tail
• C. If the person is of Type 2, then the result is tail
• D. If the person is of Type 1, then the result is tail

Answer 1

The correct option is A The result if head

Type 1 always tells trruth.
Type 2 always tells lies.
The statement is "toss is head if and only if I am telling the truth".
There are two cases possible.
Case-1: Let the person be type 1.
Case-2: Let the person be type 2.
In each we shall prove that result is head.
Case-1: Let the person be type 1. Type 1 always tells truth.
So the statement "toss is head if and only if I am telling the truth" is true .
So toss head $\iff$ telling truth. Since type 1 is telling truth so toss head is also true. So in case 1 result is that toss is head.
Case-2 : Let the person be type 2. Type 2 always tells lies.
So the statement "toss is head if and only if I am telling the truth" is false.
So toss head $\iff$ telling truth is false. So toss head $⨁$ telling truth. So toss head and telling truth have opposite truth values. Now, since type 2 telling truth is false, so toss head has to have opposite truth value which is true.
So toss head is true .So in case 2 also, result is head.
So in both cases we have proved that the result is head.
So option (a) is correct.