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I read the following passage in Barabasi - Bursts (http://www.amazon.com/Bursts-Hidden-Pattern-Behind-Everything/dp/0525951601)

A 12 member jury make a correct verdict 80 percent of the time and an incorrect 20 percent of the time.

P verdict | guilty: .8

P verdict | not guilty: .2

The probability of all juries making an incorrect verdict is therefore $.2^{12} = 0.000000004$.

I also know that the outcome preferred by the majority of jurys, before the jury discussed the case coincided with the final verdict 91 percent of the time.

Now what I don't understand is the following:

Therefore, to calculate the outcome of a verdict it is sufficient to consider the view of the majority. We can adjust our above calculation to do just that, and now the probability that the twelve-member jury will wrongly convict an innocent defendant jumps from 0.000000004 to 0.4 percent

I'm trying to figure out how this calculation was 'adjusted' but I can't.

Can someone help?

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    Don't jury decisions have to be unanimous? Is a majority vote on the jury sufficient for a verdict? I am not sure that "Therefore, to calculate the outcome of a verdict it is sufficient to consider the view of the majority." makes any sense.2011-10-24

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The simple mathematical answer to your question is

$0.2^{12} = 0.000000004096$

$\sum_{n=7}^{12} {12 \choose n} 0.2^n 0.8^{12-n} = 0.003903131648$

and this latter figure is about 0.4%.

If $N$ is the number of jurors who are wrong then the probabilities that $N=n$ are:

n   Probability 0   0.068719476736 1   0.206158430208 2   0.283467841536 3   0.236223201280 4   0.132875550720 5   0.053150220288 6   0.015502147584 7   0.003321888768 8   0.000519045120 9   0.000057671680 10  0.000004325376 11  0.000000196608 12  0.000000004096 

Just add up the last six of these.

In reality I would expect a wider spread as I would doubt juror error was independent: all 12 have seen the same potentially misleading evidence.

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    @DBR: Percentages are per hundred. So something that happens once in a hundred times is 1%. Something that happens 4 times in 1000 is 0.4%2011-10-23