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## A stranger is a strange land

Posted by Chris on February 17, 2011 – 11:03 am

You are a stranger in a strange land. All the people there tell the truth only 1/4 of the time.

You meet a couple of dignitaries. The first extols the wonders of their land. The second said that the first had spoken truthfully.

What is the probability that the first had actually spoken truthfully?

This post is under “Logic” and has 15 respond so far.

### 15 Responds so far- Add one»

1. 1. BearSprite Said：

1/4. Regardless of what the second person says.

2. 2. b Said：

1/4

3. 3. Chris Said：

Hi BearSprite and b. That would be correct if and only if the second dignitary’s statement was a wild shot in the dark that by pure luck (with a very low probability) seemed to be a response to the first dignitaries speech. i.e. as if the second dignitary hadn’t heard what the first had said. I’m meaning something along the lines of a bunch of monkeys knocking out a Shakespeare play.

In fact, let’s assume that that could have happened. Let the probability of that happening be p. Then on p occassions, the probability that dignitary 1 spoke the truth is 1/4, and on 1-p occassion, as before, the probability that dignitary 1 spoke the truth is 1/10. Altogether that gives p/4 + (1-p)/10 = 1/10 – 3p/20. As p → 0, the overall probability → 1/10.

The problem assumes that the second dignatary’s statement was directly related to what the first dignitary had actually said.

4. 4. Nathan Said：

1/10

(0.25*0.25)/(0.25*0.25+0.75*0.75)=1/10

This is another problem in conditional probability

5. 5. cheennzzzz Said：

For the first person to answer correctly the probability is 1/4 as we know and for the second person to say it right too… it will be 1/4.
Therefore… 1/4 * 1/4 = 1/16.

6. 6. Chris Said：

Well done Nathan . Would you like to expand on the logic a bit (as this is a little more involved than straightforward conditional probability)?

cheennzzzz. You haven’t considered that they might both be lying: the probability of that would be 3/4*3/4 = 9/16. Because of what the second said, they must both be lying or both telling the truth. But 1/16 + 9/16 = 10/16 < 1. What do you associate with the missing 6/16 probability?

7. 7. Nathan Said：

If the first dignitary tells the truth, then the only way the second could say the first spoke the truth would be to speak the truth himself. If the first dignitary lies, then the second must also be lying if he claims the first told the truth. The probability of these events are (1/4)^2 and (3/4)^2 respectively. These two answers constitute the universe under consideration, and the sum of their values give the denominator for our calculation, with (1/4)^2 as the numerator where the first dignitary spoke truthfully. This yield the calculation (1/4)^2/((1/4)^2)+(3/4)^2) = (1/16)/(10/16) = 1/10

8. 8. Chris Said：

Thanks Nathan. Yep, it isn’t possible for one to have told the truth and for one to have lied.

At least nobody said 1/2 (yet)

9. 9. Arsheen Said：

2/4

10. 10. Dual Aspect Said：

I hope Arsheen is pulling Chris’s leg with that one!

11. 11. Chris Said：

LOL. Good one Arsheen.

12. 12. anthony Said：

1/4*1/4/2=1/32

13. 13. Chris Said：

Hi anthony. The expression you wrote isn’t defined. 1/4/2 could be interpreted as (1/4)/2 = 1/8 or 1/(4/2) = 1/2. I deduce that you meant the former.

Nathan has explained why the answer is 1/10.

14. 14. Tom Said：

Nathan, your answer only holds if the question had been the probability of the second having having said the truth. The question asks the probability of the first person having said the truth which is independent of the second persons response and therefore still carries the 1/4 probability otherwise the defining statement in the problem (that ‘all’ the people tell truth only ‘1/4′ of the time) loses validity.

15. 15. Chris Said：

Tom. Nathan has answered correctly. You must take all of the facts provided into consideration.

You would be right if the second person’s statement didn’t refer to the first person’s statement. That fact that the second agreed with the first makes it more likely that the first had lied.

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