Find the probability that person $A$ told the truth given that person $D$ told the truth

Question

Person $A$ get's an information that he tells to person $B$ and then person $B$ tells that information to person $C$ and so on.. Given that a person only says the truth in $\frac{1}{3}$ of the cases, what is the probability that person $A$ told the truth if person $D$ told the truth? The information that is being spread is one bit long and can be $\frac{0}{1}$.

Answer 1

Let's represent a string like $TFT$ to mean that A was telling the truth, $B$ was not telling the truth, and $C$ was telling the truth. So if they all tell the truth, it would be $TTT$, and if they all lie, it's $FFF$

Now, in which cases does $D$ get the correct message? When there is an even number of lies, because given that the information is just 1 bit, two lies will get the information back to the right bit. So this is in the cases of $TTT$, $FFT$, $FTF$, and $TFF$.

The probabilities of these are:

$P(TTT) = (\frac{1}{3})^3 = \frac{1}{27}$

$P(FFT) = P(FTF) = P(TFF) = (\frac{2}{3})^2*(\frac{1}{3}) = \frac{4}{27}$

Note that $A$ was telling the truth in the case of $TTT$ and $TFF$ . So, the probability that $A$ was telling the truth if $D$ got the truth is

$$\frac{P(TTT)+P(TFF)}{P(TTT) + P(FFT) + P(FTF) + P(TFF)} = \frac{\frac{1}{27} + \frac{4}{27}}{\frac{1}{37}+\frac{4}{27} +\frac{4}{27}+\frac{4}{27}} = \frac{5}{13}$$