<s>[INST] Probability of truth in a chain of statements 
Three individuals $A$,$B$ and $C$ tell the truth with probability $1/3$.
(A) $C$ makes a statement and $A$ claims that it is true. What is the probability that the statement is true.
(B) $C$ makes a statement and $A$ tells you that $B$ claims the statement is true. What is the probability that the statement is true.

Let $T_A$ be the event that $A$ tells the truth and similarly $T_B \ \& \ T_C$
My answer in the first case $\frac{1}{3}\cdot \frac{1}{3} = \frac{1}{9}$.

I am confused with part $B$ of the question.
My reasoning: Probability of $C$ telling the truth is $1/3$. Now we don't know what $B$ said so we assume two cases. Therefore we would have the probability as $P(T_A \ T_B \ T_C \cup T_A \ T_B^C \ T_C^C) = \frac{1}{3}\cdot \frac{1}{3} \cdot \frac{1}{3}+\frac{1}{3}\cdot \frac{2}{3} \cdot \frac{2}{3}$.
Is this correct? This problem appears as an exercise to Bayes theorem so I am confused as to how the theorem is applicable.
Thanks in advance for any assistance.
 [/INST] Careful. We need to distinguish between "$X$ tells the truth" and "$X$ claims that $Y$ told the truth". The former always happens with probability $1/3$, while the latter depends on whether or not $Y$ is true or false. Let's revisit part (a) and redefine our events. Define:


*

*$X_C$: $C$ makes a true statement.

*$X_A$: $A$ claims that $C$ makes a true statement.


We're asked to find the probability that $C$ made a true statement, given that $A$ claims that $C$ made a true statement:
\begin{align*}
\Pr[X_C \mid X_A]
&= \frac{\Pr[X_CX_A]}{\Pr[X_A]} \\
&= \frac{\Pr[X_CX_A]}{\Pr[X_CX_A] + \Pr[\overline{X_C}X_A]} \\
&= \frac{\frac{1 \cdot 1}{9}}{\frac{1 \cdot 1}{9} + \frac{2 \cdot 2}{9}} \\
&= \frac{1}{1 + 4} \\
&= \frac{1}{5}
\end{align*}
Can you see how to generalize this for part (b)?

EDIT: Here's part (b). Define:


*

*$X_C$: $C$ makes a true statement.

*$X_B$: $B$ claims that $C$ makes a true statement.

*$X_A$: $A$ claims that $B$ makes a true statement.


We're asked to find the probability that $C$ made a true statement, given that $A$ claims that $B$ made a true statement:
\begin{align*}
\Pr[X_C \mid X_A]
&= \frac{\Pr[X_CX_A]}{\Pr[X_A]} \\
&= \frac{\Pr[X_CX_BX_A] + \Pr[X_C\overline{X_B}X_A]}{\Pr[X_CX_BX_A] + \Pr[X_C\overline{X_B}X_A] + \Pr[\overline{X_C}X_BX_A] + \Pr[\overline{X_C} \, \overline{X_B}X_A]} \\
&= \frac{\frac{1 \cdot 1 \cdot 1}{27} + \frac{1 \cdot 2 \cdot 2}{27}}{\frac{1 \cdot 1 \cdot 1}{27} + \frac{1 \cdot 2 \cdot 2}{27} + \frac{2 \cdot 2 \cdot 1}{27} + \frac{2 \cdot 1 \cdot 2}{27}} \\
&= \frac{1 + 4}{1 + 4 + 4 + 4} \\
&= \frac{5}{13}
\end{align*}
This diagram might help:

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