Probability

[vishwas.arora]

Problem:

A man is known to speak truth 1 out of 3 times. He throws a die and reports that it is a three. What is the probability that it is actually a three?

Request an expert to please give a brief explanation.

Thanks

[cans]

out of 3 times, he speaks truth once. thus prob = 1/3.
p(3/reported = 3) = p(3 and reported=3)/p(reported = 3) = (1/6)*(1/3)/(1/3) = 1/6

[vishwas.arora]

out of 3 times, he speaks truth once. thus prob = 1/3.
p(3/reported = 3) = p(3 and reported=3)/p(reported = 3) = (1/6)*(1/3)/(1/3) = 1/6

This is, indeed, not the correct answer.

The correct answer is [spoiler]1/11[/spoiler]

Experts pl help..!

[Ian Stewart]

Problem:

A man is known to speak truth 1 out of 3 times. He throws a die and reports that it is a three. What is the probability that it is actually a three?

Request an expert to please give a brief explanation.

Thanks

The answer is 1/11 only if you make a pretty ridiculous assumption - that when he lies, he always says "I rolled a 3". If that's the case, then there's a 1/6 chance he will roll a 3, and a 1/3 chance he will tell the truth about it, so a (1/3)(1/6) = 1/18 chance he both rolls a 3 and tells the truth. There's a 5/6 chance he does not roll a 3, and a 2/3 chance he lies and says he rolled a 3, so a (5/6)(2/3) = 10/18 chance he claims he rolled a 3 but did not. So when he claims he rolled a 3, the ratio of times he's telling the truth to the times he's lying is 1 to 10 (looking at the numerators of the two fractions), and the probability he actually rolled a 3 is 1/11.

But there's just no reason to think here that when this man lies, he will always say he rolled a 3. Why wouldn't he sometimes lie and say he rolled a 2, or a 6? If he simply chooses some other number on the die when he lies, then the probability he's telling the truth would simply be 1/3.

And of course, I'm not even sure why we'd assume here that this man tells plausible lies. How do we know that when this man lies, he doesn't say "I rolled a 29", or "What are you talking about, I didn't roll the die at all!".

I've seen a few questions set up like the one in the post above, and they're all terribly written. Unless the question defines precisely what this man does when he lies, there is just no way to answer it.

[pemdas]

a die of six numbers from 1 to 6 is reported to be truthful number 1 out of 3 times

a man throws a die and reports three; when he reports three, there is 1/6 chance that it's three. But even with the landed number three he may report not three 2/3 times, and only 1/3 times he will report three. Hence the chance that a man reports number three correctly is 1/6 * 1/3 or 1/18

Problem:

A man is known to speak truth 1 out of 3 times. He throws a die and reports that it is a three. What is the probability that it is actually a three?

Request an expert to please give a brief explanation.

Thanks

[vishwas.arora]

Problem:

A man is known to speak truth 1 out of 3 times. He throws a die and reports that it is a three. What is the probability that it is actually a three?

Request an expert to please give a brief explanation.

Thanks

The answer is 1/11 only if you make a pretty ridiculous assumption - that when he lies, he always says "I rolled a 3". If that's the case, then there's a 1/6 chance he will roll a 3, and a 1/3 chance he will tell the truth about it, so a (1/3)(1/6) = 1/18 chance he both rolls a 3 and tells the truth. There's a 5/6 chance he does not roll a 3, and a 2/3 chance he lies and says he rolled a 3, so a (5/6)(2/3) = 10/18 chance he claims he rolled a 3 but did not. So when he claims he rolled a 3, the ratio of times he's telling the truth to the times he's lying is 1 to 10 (looking at the numerators of the two fractions), and the probability he actually rolled a 3 is 1/11.

But there's just no reason to think here that when this man lies, he will always say he rolled a 3. Why wouldn't he sometimes lie and say he rolled a 2, or a 6? If he simply chooses some other number on the die when he lies, then the probability he's telling the truth would simply be 1/3.

And of course, I'm not even sure why we'd assume here that this man tells plausible lies. How do we know that when this man lies, he doesn't say "I rolled a 29", or "What are you talking about, I didn't roll the die at all!".

I've seen a few questions set up like the one in the post above, and they're all terribly written. Unless the question defines precisely what this man does when he lies, there is just no way to answer it.

Makes sense, Ian. Thanks a ton for sharing your thoughts.

Regards