Manager (variation)

Probability Level 3

In an office with 4 people (Alice, Bob, Carol, David with first initials A, B, C, D),

  • the manager lies most of the time but speaks the truth occasionally;
  • the engineers speak the truth most of the time but lie occasionally.

Given this scenario, order the four people from most likely to least likely manager.

Alice: "Bob is an engineer."
Bob: "Alice is an engineer."

Carol: "David is the manager."
David: "Carol is the manager."

Alice: "David is the manager."
David: "Alice is the manager."


Details and Assumptions:

  • The probability of the manager lying is equal to the probability of an engineer speaking the truth.
  • The probability of a statement being true or false does not depend on other statements.

Bonus question : If the first statement under "Details and Assumptions" is removed, might that possibly affect the answer to this problem?

Inspiration

D > A > B > C D > A > C > B D > C > B > A D > C > A > B

Arjen Vreugdenhil by Arjen Vreugdenhil , 44, USA

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1 solution

Arjen Vreugdenhil
Nov 16, 2017

Let p > 1 2 p > \tfrac12 be the probability of the manager lying = the probability of the engineer speaking the truth, and q = 1 − p q = 1 - p . In the table below, entries in boldface occur with probability p p , i.e. are the more likely ones.

manager A B C D A: "B is engineer" true false t r u e t r u e B: "A is engineer" false true t r u e t r u e C: "D is manager" false false f a l s e t r u e D: "C is manager" false false t r u e f a l s e A: "D is manager" f a l s e false false t r u e D: "A is manager" t r u e false false f a l s e probability p 2 q 4 q 6 p 4 q 2 p 6 \begin{array}{r|cccc} \text{manager} & A & B & C & D \\ \hline \text{A: "B is engineer"} & \text{true} & \text{false} & \mathbf{true} & \mathbf{true} \\ \text{B: "A is engineer"} & \text{false} & \text{true} & \mathbf{true} & \mathbf{true} \\ \text{C: "D is manager"} & \text{false} & \text{false} & \mathbf{false} & \mathbf{true} \\ \text{D: "C is manager"} & \text{false} & \text{false} & \mathbf{true} & \mathbf{false} \\ \text{A: "D is manager"} & \mathbf{false} & \text{false} & \text{false} & \mathbf{true} \\ \text{D: "A is manager"} & \mathbf{true} & \text{false} & \text{false} & \mathbf{false} \\ \hline \text{probability} & p^2q^4 & q^6 & p^4q^2 & p^6 \end{array}

The columns with more statements in boldface, or more factors p p , are the more likely scenario. Thus the order is D > C > A > B \boxed{D > C > A > B} .

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