One percent of the people in Westeros belong to the secret Brotherhood society. The Chief Interrogator...

One percent of the people in Westeros belong to the secret Brotherhood society. The Chief Interrogator of Westeros (“the tickler”) has a 95% chance of ruling that an interrogated person belongs to the Brotherhood, when indeed this person is one of its members. For people who are not members, “the tickler” deems them trustworthy (i.e., not a member of the Brotherhood) with probability 0.4. Suppose “the tickler” interrogates a random person from Westeros and finds them guilty of belonging to the Brotherhood. What is the probability that this person indeed belongs to the Brotherhood?

Solutions

Expert Solution

We are given,

P(actually belong to Brotherhood) = 1% = 0.01

P(actually dont belong to Brotherhood) =1 - 0.01 = 0.99

P(Tickler says belong to Brotherhood | actually belong to Brotherhood) = 95% = 0.95

P(Tickler says dont belong to Brotherhood | actually belong to Brotherhood) = 1 - 0.95 = 0.05

P(Tickler says dont belong to Brotherhood | actually dont belong to Brotherhood ) = 0.4

P(Tickler says belong to Brotherhood | actually dont belong to Brotherhood ) = 1 - 0.4 = 0.6

We need to find :

Probability that the person indeed belongs to the Brotherhood given that Tickler says he belong to Brotherhood

= P(Actually belongs to brotherhood and Tickler says he belong to Brotherhood) / P(Tickler says he belong to Brotherhood)

Now,

P(Tickler says he belong to Brotherhood) =

= P(actually belong to Brotherhood)* P(Tickler says belong to Brotherhood | actually belong to Brotherhood) +

P(actually dont belong to Brotherhood) * P(Tickler says belong to Brotherhood | actually dont belong to Brotherhood )

= 0.01* 0.95 + 0.99*0.6

= 0.6035

P(Actually belongs to brotherhood and Tickler says he belong to Brotherhood) =

= P(actually belong to Brotherhood)* P(Tickler says belong to Brotherhood | actually belong to Brotherhood)

= 0.01*0.95 = 0.0095

Probability that the person indeed belongs to the Brotherhood given that Tickler says he belong to Brotherhood =

= 0.0095 / 0.6035

= 0.0157

orchestra answered 1 year ago

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