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The Prosecutor's Fallacy & Fine Tuning

Envisage
Posts: 3,646
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3/9/2015 7:51:38 AM
Posted: 1 year ago
Prosecutor's Fallacy
A really good case-study is the Sally-Clark case, which really illustrates the issue all the way until modern court cases. Essentially the case is as follows:

Sally Clark Case
Sally Clark had two children, both consecutively died from "cot-death", she was prosecuted for murder and convicted in 1999, largely based on the statistical evidence presented.

Essentially, the background rate of natural cot-deaths per infant per person was 1 in 8500, thus for each 8,500 children born (of Sally Clark's socioeconomical standing) only one will suffer a cot-death.

To extend to two-person families, the prosecution came to the number of 1 in 73,000,000 that both children in a two-person family would die of a cot death. The number was astonishing and she was prosecuted.

There are numerous problems with this statistical analysis (such as the absurd assumption of mutually exclusive probabilities, etc), but for this thread I will concentrate on the sharp-shooter's fallacy.

Essentially, there is a dichotomy, either both children dies of cot-deaths, or they did not. If not then we can assume for the sake of argument they were indeed murdered (it's false, but I am assuming this for this thread). The problem here is there is that there is no way to determine the probability "given we have two dead children, what are the odds of them not dying of cot-deaths", since the "given we have two dead children" is the biased qualifier. We *already have* two dead children, thus the statistics that are based "per children born" are completely irrelevant. Since we are only ever going to be be selecting for trial the families with two dead children in them.

To give a more ambiguous example, if we have 6000 people roll a fair dice, and then accuse only those who roll a six of cheating, then we have a problem. Since we may well get 1000 people who rolled a six, however the 1 in 6 probability is based on "per rolled dice" rather than "per rolled dice that landed on a six", the very act of selecting people who have rolled a six sided dice renders the "one in six" statistic irrelevant in determining whether the dice was genuinely rolled, or was not genuinely rolled (e.g. via. cheating).

Extending back to the Sally Clark case, the very fact she is in the prosecutors dock based on the two murders renders the "per birth" statistic irrelevant. It may well be the case that by sheer bad luck that Sally Clark lost her children, but without other statistics (i.e. the odds of both children dying of a cot death given that both children are dead) it's impossible. One would need to compare the rate of genuine cot deaths to the rate of deaths overall in similar circumstances.

If for example, there are 1,000 cot deaths of infants against a total of 1,100 deaths overall of infants, and assuming that the 100 extra deaths are all murders (absurd, yes, but assumed for this thread), then more meaningful statistics can be concluded.

To give one extreme example, I dealt myself a 52-card hand below at random:
"http://www.debate.org... /4/3382/132431-3382-p3u9b-a.jpg"

The odds of getting this hand at random is an astonishing one in 8x10^67 (ten with sixty seven 0's after it). To claim anything else is at work however would require additional statistics outside of the probability of getting this hand out of random. The question that needs to be asked is "given this hand, what are the chances of it being from a random selection". This is basic conditional probability that anyone who does a basic statistics course will learn.[http://en.wikipedia.org...]

Thus, we come to the absurd mathematical fine-tuning arguments, which employs exactly this fallacy ad nauseum, there is simply no other way to put it. Calculating probabilities of "this" universe occurring at random says absolutely nothing about the question at hand:
"Given this universe, what is the probability of it occurring randomly"

Ignoring the key conditional here and charging on with these absurd arguments tells me one of two things:
1. You have not studied basic statistics, or flunked the class
Or.
2. You are being intentionally dishonest

Discuss. Discussions not directly about the prosecutor's fallacy's application to the fine tuning argument will be regarded as derailing the topic, I want this discussion to remain narrow.
dee-em
Posts: 6,444
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3/9/2015 9:31:23 AM
Posted: 1 year ago
At 3/9/2015 7:51:38 AM, Envisage wrote:
Prosecutor's Fallacy
A really good case-study is the Sally-Clark case, which really illustrates the issue all the way until modern court cases. Essentially the case is as follows:

Sally Clark Case
Sally Clark had two children, both consecutively died from "cot-death", she was prosecuted for murder and convicted in 1999, largely based on the statistical evidence presented.

Essentially, the background rate of natural cot-deaths per infant per person was 1 in 8500, thus for each 8,500 children born (of Sally Clark's socioeconomical standing) only one will suffer a cot-death.

To extend to two-person families, the prosecution came to the number of 1 in 73,000,000 that both children in a two-person family would die of a cot death. The number was astonishing and she was prosecuted.

There are numerous problems with this statistical analysis (such as the absurd assumption of mutually exclusive probabilities, etc), but for this thread I will concentrate on the sharp-shooter's fallacy.

Essentially, there is a dichotomy, either both children dies of cot-deaths, or they did not. If not then we can assume for the sake of argument they were indeed murdered (it's false, but I am assuming this for this thread). The problem here is there is that there is no way to determine the probability "given we have two dead children, what are the odds of them not dying of cot-deaths", since the "given we have two dead children" is the biased qualifier. We *already have* two dead children, thus the statistics that are based "per children born" are completely irrelevant. Since we are only ever going to be be selecting for trial the families with two dead children in them.

To give a more ambiguous example, if we have 6000 people roll a fair dice, and then accuse only those who roll a six of cheating, then we have a problem. Since we may well get 1000 people who rolled a six, however the 1 in 6 probability is based on "per rolled dice" rather than "per rolled dice that landed on a six", the very act of selecting people who have rolled a six sided dice renders the "one in six" statistic irrelevant in determining whether the dice was genuinely rolled, or was not genuinely rolled (e.g. via. cheating).

Extending back to the Sally Clark case, the very fact she is in the prosecutors dock based on the two murders renders the "per birth" statistic irrelevant. It may well be the case that by sheer bad luck that Sally Clark lost her children, but without other statistics (i.e. the odds of both children dying of a cot death given that both children are dead) it's impossible. One would need to compare the rate of genuine cot deaths to the rate of deaths overall in similar circumstances.

I don't see the fallacy I'm afraid. The probability calculations made by the prosecutor are quite valid in themselves. (I would have raised objections as the defense in relation to possible genetic predisposition, sleeping and feeding practices, clothing, etc. which is where your issue with mutually exclusive probabilities may come in).

If for example, there are 1,000 cot deaths of infants against a total of 1,100 deaths overall of infants, and assuming that the 100 extra deaths are all murders (absurd, yes, but assumed for this thread), then more meaningful statistics can be concluded.

Huh?

To give one extreme example, I dealt myself a 52-card hand below at random:
"http://www.debate.org... /4/3382/132431-3382-p3u9b-a.jpg"

The odds of getting this hand at random is an astonishing one in 8x10^67 (ten with sixty seven 0's after it). To claim anything else is at work however would require additional statistics outside of the probability of getting this hand out of random. The question that needs to be asked is "given this hand, what are the chances of it being from a random selection". This is basic conditional probability that anyone who does a basic statistics course will learn.[http://en.wikipedia.org...]

Yes, that is valid, you had to get some hand. You can't look at that hand, after the fact, and say it was for all practical purposes impossible. However, this isn't the situation with the prosecutor example above where there is uncertainty about whether it was really cot-death or murder. You're comparing cards (known fixed outcomes) with child deaths where you talk about cot-death versus life (1 in 8500) but completely discount the issue of deliberate murder. It's apples and oranges. The prosecution was quite right to point to the long odds of a double cot-death. The only way you could discredit this argument was if you knew for certain that it was cot-death. It's the uncertainty that undoes your reasoning.

Thus, we come to the absurd mathematical fine-tuning arguments, which employs exactly this fallacy ad nauseum, there is simply no other way to put it. Calculating probabilities of "this" universe occurring at random says absolutely nothing about the question at hand:
"Given this universe, what is the probability of it occurring randomly"

Yes, this corresponds to your dealt hand example. The universe exists. There is no point calculating probabilities (which are uneducated guesses anyway) for something which exists. Since it exists, the universe is a certainty regardless of how low the probability might be of it happening by chance. A better example might be you yourself as the unique individual you are. Think about all those tens of thousands of generations of men and women who had to meet at exactly the right time, reproduce with a particular sperm and egg, and mingle their DNA just so in order to produce you. The odds must be staggering.

Ignoring the key conditional here and charging on with these absurd arguments tells me one of two things:
1. You have not studied basic statistics, or flunked the class
Or.
2. You are being intentionally dishonest

Discuss. Discussions not directly about the prosecutor's fallacy's application to the fine tuning argument will be regarded as derailing the topic, I want this discussion to remain narrow.
Wylted
Posts: 21,167
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3/9/2015 10:07:38 AM
Posted: 1 year ago
The thing about the sharp shooter fallacy is that it looks like legitimate arguments to most people. They don't understand that everything is a 1 in a million chance when you look in reverse. The chances of me wearing my underwear brown side out, two different colored socks, stepping on a cat, and checking my mail at precisely 7:45:36 AM is about 1 in 20 million.

Stating the odds makes that stuff seem truly miraculous.
Wylted
Posts: 21,167
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3/9/2015 10:09:38 AM
Posted: 1 year ago
The sharp shooter fallacy is probably why everybody in America thinks the evidence against Jodi Arias is staggering.