s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells you: "$B$ observed $A$ truthfully" -- meaning $C$ claims that $B
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s report about $A$ was accurate.

Given $C

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement, what is the probability that $A$ is actually telling the truth?

Hints

  1. Think of each person as a noisy channel: truth preserves the signal, lying flips it. Two flips cancel out.
  2. The chain of $B$ and $C$ preserves the message with probability $(1/3)^2 + (2/3)^2 = 5/9$. Use this to compute the likelihood of $C
    s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
    Loading problems...
    s statement literally: $C$ asserts that *$B$ relayed $A
    Loading problems...
    s claim accurately* -- i.e. $C$ is making a claim about $B
    Loading problems...
    s fidelity, not about the content of $A
    Loading problems...
    s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
    Loading problems...
    s statement carry any information about $A
    Loading problems...
    s truthfulness at all? Whether $A$ told the truth is decided by $A
    Loading problems...
    s own coin flip; $C
    Loading problems...
    s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
    Loading problems...
    s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
    Loading problems...
    s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
    Loading problems...
    s output under each hypothesis about $A$.
  3. Apply Bayes' theorem: the posterior is $P(A \text{ truth} \mid \text{output}) = \frac{(1/3)(5/9)}{(1/3)(5/9) + (2/3)(4/9)} = \frac{5}{13}$.

Worked Solution

How to Think About It: This looks like a "telephone game" with liars, but the trap is in what $C$ actually says. Read $C

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s fidelity, not about the content of $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement carry any information about $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s own coin flip; $C
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$.

Quick Estimate: The prior that $A$ is truthful is

/3$. Since $C$ is talking about $B$, not about $A$, learning $C
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior,
/3$.

Approach: Identify exactly which random variables $C

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement depends on, show it is independent of the event "$A$ is truthful," and conclude that the posterior equals the prior.

Formal Solution:

Define three independent indicator events, each determined by one person's independent truth/lie coin: - $T_A$ = "$A$ tells the truth," with $P(T_A) = \tfrac13$ (and $P(\text{$A$ lies}) = \tfrac23$). - $F_B$ = "$B$ reports $A

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim faithfully (accurately)." $B$ relays faithfully exactly when $B$ tells the truth, so $P(F_B) = \tfrac13$. - $T_C$ = "$C$ tells the truth," with $P(T_C) = \tfrac13$.

These three are mutually independent because each person's truth/lie behavior is independent of the others.

Step 1: Express $C

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s statement. $C$ tells us "$B$ observed $A$ truthfully," i.e. $C$ claims that $F_B$ holds. $C
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s spoken statement $S$ = "$B$ was faithful" is *true as spoken* when $C$ is truthful and $B$ really was faithful, or when $C$ lies and $B$ really was not faithful. As an event, $S = (T_C \cap F_B) \;\cup\; (\lnot T_C \cap \lnot F_B).$ The crucial observation: $S$ is built only from $T_C$ and $F_B$. The content of $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim and the truth value $T_A$ never enter, because $C$ is asserting a fact about $B
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s *fidelity*, not re-transmitting $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim. (A faithful $B$ copies $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim verbatim whether or not that claim is true, so $B
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s fidelity says nothing about $A
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s truthfulness.)

Step 2: Independence. $T_A$ is independent of both $T_C$ and $F_B$ (the latter is determined by $B

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s coin). Hence $T_A$ is independent of any event constructed from $T_C$ and $F_B$, in particular of $S$: $P(T_A \mid S) = P(T_A).$

Step 3: Evaluate. Therefore $P(A \text{ truthful} \mid C\text{'s statement}) = P(T_A) = \frac13.$

(As a sanity check on independence: $P(S) = P(T_C)P(F_B) + P(\lnot T_C)P(\lnot F_B) = \tfrac13\cdot\tfrac13 + \tfrac23\cdot\tfrac23 = \tfrac59$, and $P(T_A \cap S) = P(T_A)P(S) = \tfrac13\cdot\tfrac59 = \tfrac{5}{27}$, so $P(T_A\mid S) = \tfrac{5/27}{5/9} = \tfrac13$.)

Answer: $P(A \text{ truthful} \mid C\text{'s statement}) = \dfrac{1}{3}$.

Remark: The tempting wrong answer ($5/13$) comes from treating $C$ as the end of a noisy channel that transmits the *message* "$A$ told the truth" through $B$ then $C$, and applying Bayes with a chain-preserve probability $5/9$. But $C$ is not relaying $A

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s claim -- $C$ is commenting on whether $B$ was a faithful relay. Whether $B$ was faithful is independent of whether $A$ was truthful, so $C
s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...
s meta-statement is uninformative about $A$, and the posterior collapses back to the prior
/3$. The lesson: read the conditioning event literally before modeling it as a channel.

Intuition

Each liar in a chain acts as a binary symmetric channel with crossover probability /3$. The cascade of two such channels has a net crossover probability of $4/9$ (chain flips) and preservation probability $5/9$ (chain preserves). Even though each individual link is heavily biased toward lying, the cascade is slightly truth-preserving because two lies cancel. This is the same reason that the product of two negative numbers is positive -- flipping twice gets you back.

The Bayesian update is crucial and is where many people slip up. Hearing "$A$ told the truth" from the chain is more likely when $A$ actually lied (because $P(\text{lie}) = 2/3$ is high, and the chain-flip probability $4/9$ is not negligible). So the posterior on $A$ being truthful, $5/13 \approx 0.385$, is only modestly above the prior of

/3$. The chain barely helps us learn about $A$.

Open the full interactive solver →

s claim and reports it to $C$ (either faithfully or with a lie). $C$ then tells yo…","eduQuestionType":"Flashcard","url":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html","acceptedAnswer":{"@type":"Answer","text":"How to Think About It: This looks like a \"telephone game\" with liars, but the trap is in what $C$ actually says. Read $C
Loading problems...
s statement literally: $C$ asserts that *$B$ relayed $A
Loading problems...
s claim accurately* -- i.e. $C$ is making a claim about $B
Loading problems...
s fidelity, not about the content of $A
Loading problems...
s claim or about whether $A$ was truthful. So before grinding through a noisy-channel Bayes computation, ask: does $C
Loading problems...
s statement carry any information about $A
Loading problems...
s truthfulness at all? Whether $A$ told the truth is decided by $A
Loading problems...
s own coin flip; $C
Loading problems...
s statement is a function only of (i) whether $B$ was faithful and (ii) whether $C$ lied. Those are independent of $A$. Quick Estimate: The prior that $A$ is truthful is $1/3$. Since $C$ is talking about $B$, not about $A$, learning $C
Loading problems...
s statement should not move that prior. Estimate: the answer is exactly the prior, $1/3$. Approach: Identify exactly which random variables $C
Loading problems...
s statement depends on, show it is independent of the event \"$A$ is truthful,\" and conclude that the posterior equals the prior. Formal Solution: Define three independent indicator event…"}},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://quantvault.org/"},{"@type":"ListItem","position":2,"name":"Problems","item":"https://quantvault.org/problems.html"},{"@type":"ListItem","position":3,"name":"Probability","item":"https://quantvault.org/probability-interview-questions.html"},{"@type":"ListItem","position":4,"name":"Truth Through a Chain of Liars","item":"https://quantvault.org/problem-659-truth-through-a-chain-of-liars.html"}]}]}
Loading problems...