Question : If X speaks the truth in 65% and Y in 75% cases. If they speak a fact independently, find the following probabilities

(i)                                    
X speaks the truth but Y tells a lie

(ii)                                 
X tells a lie but Y speaks the truth

(iii)                               
Both speak the truth

(iv)                              
Both tell a lie

(v)                                 
they contradicts each others

Answer :

Given that,

X speaks the truth in 65% and Y in 75%  cases.

So the probabilities of speaking truth and telling a lie are

P(X speaks truth)=65% =0.65,

P(X tells a lie)=100%-65% =35%=0.35,

P(Y speaks truth)=75% =0.75,

and P(Y tells a lie)=100%-75% =25%=0.25

If they speak a fact independently then the probability that
both speak the truth is

(i)                              
The probability that X speaks the truth but Y
tells a lie

P(X speaks truth but Y tells a lie)= P(X speaks truth)*P( Y
tells a lie)=0.65*0.25 =0.1625

(ii)                         
The probability that X tells a lie but Y speaks
the truth

P(X tells a lie but Y speaks the truth)= P(X tells a lie)*P(
Y speaks the truth)= 0.35*0.75=0.2625

(iii)                    
The probability that  Both speak the truth

P(Both speak the truth)= P(X speaks the truth)*P( Y speaks
the truth)=0.65*0.75= 0.4875

(iv)                     
The probability that  Both tell a lie

P(Both tell a lie)= P(X tell a lie)*P( Y tell a lie)= 0.35*0.25=0.0875

(v)                          
The probability that they contradicts each others

They can contradict each others when
X speaks truth and Y tells a lie  or X tells
a lie false and Y speaks truth.

Therefore the probability that they
contradict each others is

P(They contradicts each others) =
P{( X speaks truth and Y tells a lie  )}+P{(
X tells a lie false and Y speaks truth)}

=0.65*0.25 +0.35*0.75 =0.425