I have issues with the answer for this question problem:
Clarence is known to speak the truth two out of three times. He throws a fair, six-sided die and reports that it shows a 6. What is the probability that the number shown on the die is really 6?
I will explain my issues on this problem, forgive me if I use the wrong terminologies as I’m not that good with them myself.
The first instinctive method that comes to me would be to apply Bayes’ Theorem because there is an event(positioned at the end of the probability tree) that is observed, and the questions asks for the probability that this observation is conditioned on another observation(in the early part of the tree). Here's my Tree:
From the way I drew my tree, denote A as P(Rolls 6 and says 6)= 1/6*2/3=1/9 Denote B as P(rolls non-6 and says 6) = 5/6*1/3*1/5=1/18. Required probability = A/(A+B) = 2/3.
However, I saw some fellow classmates attempt the problem by considering B to be 5/6*1/3 instead, causing required probability to be 2/7 instead.
I am not aware of the correct solution for this as I don’t know the full assumptions of Bayes’ Theorem.
Under my assumption, there seems to be a discrepancy because I do not have a uniformly branched tree, i.e I only actually considered the 3rd level of branch for Lying and sticked to the 2nd for Truth, because I force the Truth to be said = actual, hence P(said 6|Truth and rolled 6)=1, which is unlike most of the other problems I have encountered while using Bayes.
Under my classmates assumption, they assume P(say 6|lie and roll 6) = 1, which doesn’t make much sense either, because in reality, a lie is a mismatch between “said” and “actual”, which conflicts with this questions assumption that there is a observed P(Truth)=2/3. Should Truth even be an independent event since it was stated as an observation? Or should Truth be dependent on the actual matching of “said” and “actual”, which also makes no sense to reduce the denominator in the probability as shown in the possibility tree.
Under both scenarios, there is a forcing of the proceeding probability to be 1, which seems like a combinatorial assumption, but doesn’t quite make sense in probability problems, as shown in the above analysis.
This which makes me wonder if there was actually a proper way to draw probability trees. In this case, consider 2 events A(truth or lie) and B(said number) and we draw them A then B from left to right. Hence we might be suggesting that in each different A1,A2…Ai, there might be different probability of B1,B2..Bi, i.e B is conditioned to A. However when B was claimed to be observed, does this contradict with our previous assumption of dependency, or do they work separately of each other? A similar argument might arise from the decision to draw B then A too.
Could someone enlighten me what should be the right way to view this question? I am very confused and it already feels strange that Truth could be denoted to be more than (1/6). Feel free to clarify on my notations used.
Side Question: My initial solution, Required P(rolled a 6|say 6) = P(Truth), is this a coincidence or is there a general context that allows this to happen?
The main problem is that there is a big difference between answering a "yes/no" question, which is deterministic (that means after deciding if to lie/tell the truth, there is only one possible answer), and being able to make statements of your own, which even after you've decided if you want to lie or tell the truth, there are many options.
If the original problem said "Clarence was asked if he threw a 6 and confirmed.", I'd agree with user21820: Your tree is 2 levels deep and your class mates got it right.
However, if Clarence could have made any statement conistent with his choice of truth/lie, then there is simply not enough information to answer the question.
You assume that if he lies, he'll pick one if the 5 incorrect options and chooses them with equal probability. Neither assumption is justified. He could lie by telling he rolled a $7$, a $3.5$ or $\pi$. He could have said he's 1000 years old, or any other lie not related to the throw.
Even if he restricts himself to answer with any of the 5 possible throws that are lies, he might prefer low numbers or high numbers.
So if Clarence is allowed to make up statements, then any analysis will need to know how he creates them, as that has consequences on the case where a number other than 6 is thrown and Clarences decides to lie.