GATE CSE 2015 Set 3 | Question: 24

Question

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?

• A. The result is head
• B. The result is tail
• C. If the person is of $\text{Type 2}$, then the result is tail
• D. If the person is of $\text{Type 1}$, then the result is tail

Comments

superb explanation

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Assume Type 1 persom is K(always Telling truth), Type 2 person is N(Always lying).

Statement is $H \leftrightarrow P(K)$ (where H means Head, P(K) means person is K i.e. telling truth )

Case 1 : Assume the person is K.

then $H \leftrightarrow P(K)$ is true, and since P(K) is true in this case, so H is also true. So, Head in this case.

Case 2 : Assume the person is N.

then $H \leftrightarrow P(K)$ is False, and since P(K) is false in this case, so H is true, so, Head in this case also.

Hence, answer A.

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is this interpretion correct that if the chosen is type-1 then he must be speaking the truth so result must be head only and no other than A contains head as option so A is answer.

Answer 1

T -> Person speaking Truth 

H-> Head appears 

Given proposition , T <=> H

1. Case 1: Person is Type 1 , speaking Truth hence HEAD.
2. Case 2: Peson is type 2 , speaking false hence negation of given proposition is what the perosn is saying ( since he is a lyer ) . In the proposition T is False ( since he is lying) , if H is false then F <=> F is True but it's a contradiction , So H has to be true to make the proposition False   .. Hence, Option A 

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Answer 2

"head if and only if he is telling the truth" 

h: head ; tr: telling the truth

$(\neg tr \lor h)\wedge(tr \lor \neg h)$

negation oof the above will give $( tr \wedge \neg h)\lor(\neg tr \wedge h)$ qhwich means :

1. Either it is not head and he is telling the truth
2. Either it is a head and he is not telling truth

The 1st one is not true as liar cannot tell the truth so 2nd one is valid.

so, answer is (A)

Answer 3

$\therefore$ In any case irrespective of who said the statement, the toss is head and answer is Option $A$

Answer 4

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 ⇔ 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 ⇔ 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.