Wikipedia:Reference desk/Archives/Mathematics/2007 September 6

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September 6

Looking for a fallacy in statistics

I'm looking for the proper name of a fallacy in statistics. It goes something like this:

You work in a factory that manufactures six-sided dice. Your factory manufactures one thousand dices a day. After each die is created, it is rolled 5 times and the results recorded.

As you are walking through the factory one day, you noticed a die with the result of five 6s in a row. You suspect that the die is "unfair" and then calculated the odds to be 1/(6^5) = 1/7776 and thus so you threw that die into the defective bin.

The fallacy is "observing the data" to form a belief and then using the exact same data to confirm that belief. What is the proper name of this fallacy?

The reason, I'm asking this is related to a thought experiment where a police notices a muslim behaving suspiciously. The muslim performed 5 suspicious activities, the police then calculated that the probability of 5 suspicious activities happening by coincidence to be 1/7776 thus the police concludes that muslim must be a terrorist.

202.168.50.40 01:17, 6 September 2007 (UTC)

Given that all dice were tested, this is a case of the Prosecutor's fallacy, specifically the multiple testing variety. Testing all as such will quite likely give a small number of unusual rolls so singling them out is tricky.

An additional problem: the nominally operative probability should be that of seeing a set of 5 identical rolls (we assume that if, for example, 5 "3"s were observed, the reaction would be the same). This is six times 1/7776 or 1/1296. This issue is roughly testing hypotheses suggested by the data, although that article could use some work. Baccyak4H (Yak!) 02:25, 6 September 2007 (UTC)

BTW. Suppose you see another die, rolled five times with a result of 1-2-3-5-6. The probability of this result is also (1/6)^5 = 1/7776. So this die should be considered "unfair", too, and land in a defective bin, together with the previous one. Am I right?
CiaPan 09:32, 6 September 2007 (UTC)

That is the probability of those rolls in that order. But I suspect any set of 5 rolls without a repitition would be so flagged as unfair. Here the operative probability is (6!/1!)/(6^6), or about 1.5%. Baccyak4H (Yak!) 13:27, 6 September 2007 (UTC)

Two further issues: 1) a better way to assess QC in this manner is to exploit the empirical observation that when there is a manufacturing problem, it tends to linger until noticed and the process fixed. Testing only a small number from the entire day's batch greatly reduces the multiple testing issue, and loses very little effectiveness (assuming a good sampling scheme). 2) The application to the thought experiment is problematic in that people generally are (and one could argue police should be) Bayesian, not frequentist (which my above replies assume). But that philosophy is subjective: one person's valid but imprudent application could end up locking up a lot of silly suspects (e.g., elderly people who become disoriented), while another's would lock up nearly every single olive-skinned person with a pulse. So there still is some ambiguity as to how such operations should work. Baccyak4H (Yak!) 13:37, 6 September 2007 (UTC)

Something a bit different, but along similar lines, would be confirmation bias. It doesn't fit the dice example very well, but it fits the other example quite a bit better, in a way I think you haven't explored yet. Black Carrot 23:45, 6 September 2007 (UTC)

One other detail on the thought experiment - there are such things, at least in America, as "reasonable doubt" and "just cause". No argument the officer gave for detaining someone on those grounds would fly. Black Carrot 18:07, 7 September 2007 (UTC)

Solving Inequalities

I'm working with deltas and epsilons and limits and I'm trying to solve an inequality. I have -0.75<1/x^2<1.25. Do I need to change the direction of the inequality when I take the reciprocal of all the numbers? And how do I do a square root when one of the numbers is negative?

Thanks! —Preceding unsigned comment added by 130.49.11.65 (talk) 02:49, 6 September 2007 (UTC)

Manipulating inequalities is much trickier than manipulating equations. With an equation, you can apply any function to both sides, and the resulting equation is still true. With inequalities, this does not work unless the function is monotonic. For example, the function f(x) = 1/x is not monotonic, so you cannot apply it to both sides of the equation, even if you change the direction:

-1 < 1 < 2

but

1/(-1) < 1/2 < 1/1

The new values are not in the same order, and they're not in the opposite order either. Their ordering has been shuffled around by the action of the non-monotonic function f(x) = 1/x.

Sometimes the best approach to solving an inequality is to find the conditions in which two of the expressions are equal, and then determine the signs of the inequalities in each of the resulting regions. And use common sense. As you know, real numbers have non-negative squares, so if x is a (nonzero) real number, then 1/x^2 >= 0 and the left part of your inequality is always satisfied. Isn't that a nice simplification? —Keenan Pepper 03:43, 6 September 2007 (UTC)

Another thing you can always do is mulptiply all sides by some positive number. You know that x must be nonzero (otherwise you can't divide by it) so $x^{2}$ is positive, so you get $-0.75x^{2}<1<1.25x^{2}$. As mentioned, only the $1<1.25x^{2}$ part actually conveys information.

You do not need to take a square root of a negative number here, since the mere usage of inequalities suggests we are dealing with real numbers, so this is impossible. If you are interested in this anyway, take a look at imaginary unit and complex number. -- Meni Rosenfeld (talk) 11:34, 6 September 2007 (UTC)