Everybody in Ourtown is a fool or a knave or possibly both. 70% of the citizens are fools and 85% are knaves. One citizen is randomly select

Question

Everybody in Ourtown is a fool or a knave or possibly both. 70% of the citizens are fools and 85% are knaves. One citizen is randomly selected to be mayor. What is the probability that the mayor is both a fool and a knave?

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Josie 2 weeks 2022-01-12T16:02:45+00:00 1 Answer 0 views 0

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    2022-01-12T16:04:24+00:00

    Answer:

    There is a 55% probability that the mayor is both a fool and a knave.

    Step-by-step explanation:

    We solve this problem building the Venn’s diagram of these probabilities.

    I am going to say that:

    A is the probability that someone in Ourtown is a fool.

    B is the probability that someone in Ourtown is a knave.

    We have that:

    A = a + (A \cap B)

    In which a is the probability that someone in ourtown is a fool but not a knave and A \cap B is the probability that someone in Ourtown is both these things

    By the same logic, we have that:

    B = b + (A \cap B)

    Everybody in Ourtown is a fool or a knave or possibly both.

    This means that the union of these sets is 1, that is,

    A \cup B = 1

    In which

    A \cup B = a + b + A \cap B

    So

    a + b + A \cap B = 1

    70% of the citizens are fools

    This means that A = 0.7

    A = a + (A \cap B)

    0.7 = a + (A \cap B)

    a = 0.7 - (A \cap B)

    85% are knaves

    This means that B = 0.85

    B = b + (A \cap B)

    0.85 = b + (A \cap B)

    b = 0.85 - (A \cap B)

    What is the probability that the mayor is both a fool and a knave?

    A mayor is a random citizen just like any other, so this probability is the same as it would be for any person.

    This is A \cap B, which we can find replacing both a and b in the equation below.

    a + b + A \cap B = 1

    0.7 - (A \cap B) + 0.85 - (A \cap B) + A \cap B = 1

    A \cap B = 0.7 + 0.85 - 1

    A \cap B = 0.55

    There is a 55% probability that the mayor is both a fool and a knave.

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