# Probability of conviction

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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$0.2^{12}$ is actually about $0.000000004$, not $0.0000004$. You also picked up an extra zero in the quotation: it should be $0.0000004$%. –  Brian M. Scott Oct 23 '11 at 20:58
The calculation $0.2^{12}$ does not match the information given. Do you mean that the probability that an individual juror arrives at an incorrect verdict is $0.2$? Then the probability that all twelve jurors do so is indeed $0.2^{12}$. –  Brian M. Scott Oct 23 '11 at 21:01
Without seeing the original, it is difficult to say what this is about. Sometimes you seem to have said "jury" when "juror" might make more sense. On top of this, I would doubt juror error was independent: all 12 have seen the same potentially misleading evidence. –  Henry Oct 23 '11 at 21:02
@BrianM.Scott yes, for the first part its individual jurors and it is 0.2ˆ12. but when looking at the majority only how does it get 0.4? –  DBR Oct 23 '11 at 21:02
@Henry yes I also think that its strange to look at juror decisions are independent, but that's how the book does its calculations, i was just reading the chapter and couldn't proceed without figuring out how 0.000000004 turned into 0.4 –  DBR Oct 23 '11 at 21:07

$$0.2^{12} = 0.000000004096$$

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

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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The 0.4% represents the chance that at least seven jurors vote incorrectly, given that each of 12 jurors vote correctly 80% of the time and the probabilities are independent. –  Ross Millikan Oct 23 '11 at 21:08
@DBR: To get a majority out of 12, you need 7 or more. –  Henry Oct 23 '11 at 21:14
It's interesting how there's more chance that 4 jurors were wrong than the chance everybody got it right! –  Asaf Karagila Oct 23 '11 at 21:17
@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% –  Ross Millikan Oct 23 '11 at 21:20