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92 votes
92 votes

In a room there are only two types of people, namely $\text{Type 1}$ and $\text{Type 2}$. $\text{Type 1}$ people always tell the truth and $\text{Type 2}$ 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?

  1. The result is head
  2. The result is tail
  3. If the person is of $\text{Type 2}$, then the result is tail
  4. If the person is of $\text{Type 1}$, then the result is tail
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12 Answers

15 votes
15 votes

A. The result is head
p : "The result of the toss is head"
q : "I am telling the truth"
    p <-> q
    
Type 1:-
    Pq + p'q' = True
Head is answer when type 1.

Type 2:-
    p <-> q = False
Note: Do not take negation on both side i.e. ~(p<->q) = True is logically incorrect because RHS part is true means Type 2 person is saying True.

    please check the truth table for Ex-NOr
                 P    Ex-NOr q      Result
    case1    F                T        F
    case2    T                F        F

    There are only 2 case when exNor becomes false
    to p <-> q become False this 2 cases both need to be false.
Case1    when ~p then q = F
                i.e. "when result is Tail then I am telling Truth" this is False statement
                i.e. "the result is Tail then I am saying Lie"
                i.e. "The result is Head"

Case2    when p then ~q = F  
                 i.e. "when result is Head then I am telling Lie" this is False statement
                 i.e."when result is Head then I am telling Truth.
                 i.e. "the result is Head"
Head is answer when Type 2.

edited by
6 votes
6 votes

Let $p=$ result of the toss is head, $q=$ I am saying the truth

$p$ $q$ $p\Leftrightarrow q$
$T$ $T$ $T$
$T$ $F$ $F$
$F$ $T$ $F$
$F$ $F$ $T$

$\textbf{Case1:}$ Type $1$ person

$p\Leftrightarrow q \equiv T$

  1. $p=T,q=T$
  2. $p=F,q=F$ , It is not possible.

Result: Head

$\textbf{Case2:}$ Type $2$ person

$p\Leftrightarrow q \equiv F$

  1. $p=T,q=F$
  2. $p=F,q=T$ , It is not possible.

Result: Head

4 votes
4 votes
whatever Type 2 person says should be false.

So if Result is head, then the statement "The result of the toss is head if and only if I am telling the truth" is false is consistent with lying.

So i think answer is A

if a person is Type 2 we know he lying -> he is not telling truth -> Result is head
4 votes
4 votes

"The result of the toss is head if and only if I am telling the truth"

say type1 people telling truth

putting negation

The result of toss is not head if and only if I am not telling truth // now is this statement false ? No it is true . right?

So, result must head as Type1 people is telling truth


Now, type2 people is telling lie

but he also telling "The result of the toss is head if and only if I am telling the truth"

Now, our question is how the same statement could be FALSE?


here is the logic

The result of the toss is head = P

I am telling the truth= Q


According to lie teller P =TRUE (means the result is true)

Q=FALSE (as FALSE teller must speaking FALSE )

Now according to the statement P⟷Q = (P⟶Q) ⋀(Q⟶P).....................(i)

Now, for being it false either P⟶ Q = FALSE or Q⟶P is FALSE. right?

Now, P⟶Q means (~P ⋁Q) i.e. The result of the toss is not head or I am telling truth = False ⋁ False = FALSE // here Q is  false as he is a lie teller

Q⟶P means(~Q⋁P) i.e. I am telling FALSE or the result of the toss is head = TRUE⋁TRUE = TRUE

Now, putting it in equation (i) we get  FALSE⋀TRUE =FALSE

So, the lier also telling same statement , but he is lying.

So,  the result is head :)

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