(logic and Bayes\'s Theorem) Here\'s a small fictitious drama with five actors:

Question

(logic and Bayes's Theorem) Here's a small fictitious drama with five actors: three people A, B and Con death row; the governor, who has chosen one of them at random to be pardoned; and a warden in the prison, who knows the identity of the person the governor picked but isn't allowed to tell A, B or C who the lucky person will be. Person A now speaks to the warden, as follows Please tell me the name of one of the other prisoners who's not going to be pardoned no harm done, since you won't be identifying the lucky person. Let's agree on these rules: if B will be pardoned, you say C; if C will get the pardon, you say B; and if I'm the lucky person, you toss a 50/50 coin to decide whether to say B or C The warden thinks it over and says "B won't get the pardon." This is good news to A, because he secretly didn't believe that the warden's statement contains no information relevant to him he thinks that, given what the warden said, his chance for the pardon has gone up froo Use Bayes's Theorem to show that A's reasoning is incorrect, thereby working out whether there was information in what the warden said that's relevant to A's probability of being pardoned. [25 points/

Explanation / Answer

Given: A small fictuous drama with five actors : Three people - A,B and C - on death row; Governor; Warden

Accoring to the rules given in question

In the above table, the first column represents the actual person who is being pardoned the next 3 columns represents the person who would be told by the governor to A that is not pardoned

Now we know that P(A)=P(B)=P(C)=1/3 initially , that is the person is not being pardoned.

Now when the governor states that B will not be pardoned , this means that either A was pardoned or C was pardoned.

P(B not pardoned | A pardoned) = 0.5 (From above table)

P(B not pardoned | C pardoned) = 1

P(B not pardoned | B pardoned) = 0

Now by using law of probability we get :

P( B not pardoned ) = P( B not pardoned | A pardoned ) P( A pardoned ) +P( B not pardoned | B pardoned ) P( B pardoned ) + P( B not pardoned | C pardoned ) P( C pardoned ) = (1/3)*(0.5+1+0) = 0.5

Now from Bayes conditional probability we get :

P(A pardoned | B not pardoned) = P (B not pardoned | A pardoned) * P(A pardoned) / P(B not pardoned)

=(1/3)*(0.5)/0.5 = (1/3)

Therefore given that B is not pardoned, the probability that A would be pardoned is still 1/3

Hence A's reasoning is still incorrect

Pardoned Person A    B C    A 0 0.5 0.5 B 0 0 1 C 0 1 0

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