In a different world, suppose a jury trial happens as follows:
There are 12 jurors that act independently of each other.
If eight or more of the jurors vote "guilty," then the jury's verdict is defendant is "guilty."
If seven or fewer of the jurors vote "guilty," then the jury's verdict is "not guilty."
Each juror makes correct decision with probability p=.7 (correct decision defined as: if defendant is guilty, juror votes guilty. if defendant is not guilty, juror votes not guilty).
On a given day, there are 10 trials where 8 of the 10 trials the defendants did commit the crime and 2 of the 10 trials the defendant is not guilty.
You attend one of the 10 trials at random, what is the probability that the jury will return a correct decision for the trial that you attend?
I tried using the sum of binomial distribution.
For a guilty person, probability they are found guilty is: Sum of (12 choose x)(.7)^x(.3)^(12-x) from 8 to 12 = .723655
For a non-guilty person, probability they are found not guilty is: Sum of (12 choose x)(.7)^x(.3)^(12-x) from 5 to 12 = .990510
Probability that the jury will return the correct decision that you attend: there are 10 trials, 8 with guilty defendants and 2 with not guilty defendants and you go to a random one so (8(.723655)+2(.990510))/10 = 0.777026
Is that the right approach or am I missing something with trying to find the probability that the jury will return the correct decision that you attend.