Wikipedia:Reference desk/Archives/Mathematics/2008 April 8

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April 8

Symmetry

Is the letter L symmetrical, diagonally?

Nick4404 yada yada yada 18:53, 8 April 2008 (UTC)

It may depend on the typeface, but in most typefaces the vertical leg of the glyph for 'L' is longer than the horizontal leg. Exaggerating this, the shape is like ⌊. What happens if you take the diagonal mirror image with respect to the diagonal, swapping horizontal with vertical? Do you get the same image back? I can't show the result directly, but if you then also rotate it a quarter turn clockwise, you get this: ⌈.  --Lambiam 20:06, 8 April 2008 (UTC)

Are testimonies useful or the terrible indicators of truth?

Here's an apparent paradox I cannot resolve. If each person in a group tells the truth 90% of the times, then P(one person telling the truth when speaking about something)=0.9, but if another person corroborates the story, P(the event actually happened per the story) falls to 0.9^2=0.81. This seems to lead to the paradox in which if many, many people confirm the story, it almost certainly did not happen. Imagine Reason (talk) 21:53, 8 April 2008 (UTC)

You have made the prosecutor's fallacy - You can only tell the posterior probability of the story being true if you know the prior, in which case you just use Bayes. If someone tells you "I have just rolled a fair dice die 100 times and they all turned up 6", you know he is lying no matter how truthful he usually is. What you have in fact calculated is the probability that they will both confirm the story given that it is true, not the other way around. -- Meni Rosenfeld (talk) 22:15, 8 April 2008 (UTC)

You also know he has poor grammar :-) --Trovatore (talk) 01:39, 9 April 2008 (UTC)

Exactly. I'm just the messenger - of course I know what the singular form is :) -- Meni Rosenfeld (talk) 15:30, 9 April 2008 (UTC)

Here is the calculation for the example. Let T denote: the story is true, F: the story is false, and CC: both people state (independently) that the story is true. Then the conditional probability P(CC | T) = 0.81. But P(CC | F) = 0.01. Now by Bayes' Theorem we have P(T | CC) = P(CC | T)P(T) / (P(CC | T)P(T) + P(CC | F)P(F)) = 81P(T) / (81P(T) + P(F)) = 81P(T) / (80P(T) + 1) – assuming that T and F are complementary events. So even if P(T) = 10% – the prior probability for a somewhat unlikely story – the posterior probability P(T | CC) that the story is true given that two people confirmed it has gone up to 90%.  --Lambiam 00:23, 9 April 2008 (UTC)

Got it. Thank you. Imagine Reason (talk) 17:47, 9 April 2008 (UTC)

Another interesting problem is that witness testimonies are not independent events. If 99 people have already said "yep, I saw him do it", then the 100th person will be far more likely to "go along with the group". If he isn't sure, he'll just trust the judgment of the earlier 100 witnesses. What's worse, even if he is certain the accused did not do what he was accused of doing, the witness would still be reluctant to contradict 99 other witnesses and be thought a liar. StuRat (talk) 04:38, 10 April 2008 (UTC)

And that would be the 'herd' mentality of humanity at work. A math-wiki (talk) 06:15, 10 April 2008 (UTC)