Probability of conviction I read the following passage in Barabasi - Bursts ( 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?

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.