Respuesta :
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:
[tex]A = a + (A \cap B)[/tex]
In which a is the probability that someone in ourtown is a fool but not a knave and [tex]A \cap B[/tex] is the probability that someone in Ourtown is both these things
By the same logic, we have that:
[tex]B = b + (A \cap B)[/tex]
Everybody in Ourtown is a fool or a knave or possibly both.
This means that the union of these sets is 1, that is,
[tex]A \cup B = 1[/tex]
In which
[tex]A \cup B = a + b + A \cap B[/tex]
So
[tex]a + b + A \cap B = 1[/tex]
70% of the citizens are fools
This means that [tex]A = 0.7[/tex]
[tex]A = a + (A \cap B)[/tex]
[tex]0.7 = a + (A \cap B)[/tex]
[tex]a = 0.7 - (A \cap B)[/tex]
85% are knaves
This means that [tex]B = 0.85[/tex]
[tex]B = b + (A \cap B)[/tex]
[tex]0.85 = b + (A \cap B)[/tex]
[tex]b = 0.85 - (A \cap B)[/tex]
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 [tex]A \cap B[/tex], which we can find replacing both a and b in the equation below.
[tex]a + b + A \cap B = 1[/tex]
[tex]0.7 - (A \cap B) + 0.85 - (A \cap B) + A \cap B = 1[/tex]
[tex]A \cap B = 0.7 + 0.85 - 1[/tex]
[tex]A \cap B = 0.55[/tex]
There is a 55% probability that the mayor is both a fool and a knave.
