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On planet XZ101X, there are two types of creatures: small brain and big brain.
Small brain tells the truth 6/7 of the time and lie only 1/7 of the time, while big brain tells the truth 1/5 of the time and lie 4/5 of the time.
It is also known that there is 2/3 chance a creature from XZ101X is a small brain and a 1/3 chance that it is a big brain but there is no way of differentiating from these two types.
You are visiting XZ101X on a research trip. During your stay, you come across a creature who states that it has found a one line proof Fermet's last theorem.
Immediately after that, a second creature shows up and states that the first creatures’ statement was a true one.
If the probability that the first creature’s statement was actually true is a/b for some co-prime positive integer a, b. The value of b-a=__________.

1 Answer

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First let us calculate these things

a) P(Small brain is there in creature)  =   2/3

   P(Big brain is there in creature)      =   1/3

  Thus these 2 are clearly mutually exclusive ..Now consequent to small brain is found , we have conditional probabilities:

b) P(Truth is spoken | small brain)     =   6/7

    P(Lie is spoken   | small brain)      =   1/7

Now considering large brain creature  :

c)   P(Truth is spoken | big brain)     =   1/5

      P(Lie is spoken | big brain)        =   4/5

Now second creature can report the first creature to be true in 2 scenarios :

a) 1st creature is actually true         =   In this case both of the creatures are true..

So P(1st creature is  true)    =  P(both creatures are true) 

                                                     =  ((2/3 * 6/7) + (1/3 * 1/5))2 [Since there are 2 creatures]

                                                     =   (4/7 + 1/15)2

                                                     =   (67 / 105)2

                                                                       =   x

Now P(1st creature is false but reported true by 2nd creature)  =  P(both are false)

                                                                                          =   ((2/3 * 1/7 + 1/3 * 4/5)2

                                                                                          =   (2/21  + 4/15)2

                                                                                                                        =   (38 / 105)2  

                                                                                          =    y

So given that creature 2 reports 1 to be true ,

P(creature 1 is actually true)                                                 =  x / x + y   =  a / b

Therefore  b - a                                                                    = y

                                                                                          = (38)2

                                                                                          =  1444

Hence  b - a         =    1444 should be the correct answer..

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