This week, one of my friends in a probability class showed me a problem about jurors and Bayes' rule. The jurors voted a guilty person innocent or voted an innocent person guilty some not-so-small fractions of the time, and a person was convicted if at least 9 jurors voted guilty. They were asked to find the probability of an correct jury decision given that some percentage of people tried were actually guilty.
This problem can be solved well with some indicator random variables (1 if a juror votes guilty, 0 otherwise), and the assumption is that they make their votes independently. In reality, however, this is generally not the case. In most (if not all) jury-based judicial systems, the jury meets and discusses the trial, making their decisions very much interdependent. So here is my question: is this desirable?
On one hand, some jurors lacking in critical thought or engagement in the trial might benefit from the knowledge and thoughts of those who are thinking about the case. On the other hand, a few "loud" jurors might heavily bias the decision merely by virtue of their persuasive abilities, regardless of the facts presented in court.
I feel that if you're going to be idealistic and follow the first train of reasoning, why would you not be equally optimistic that there simply won't be such poor jurors at all? If we are going to be realistic enough to admit that there will be jurors who neglect to use their own critical abilities, we should admit that it is more likely that a persuasive individual will influence them to follow rather than to think for themselves--in which case, their presence is not only useless but also detrimental to a fair decision.
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