## Sunday, May 17, 2015

### Jury Duty

I've been playing with some problem's from the book "50 Challenginh problems in probability with solutions by  Frederick Mosteller (my great grand advisor as it happens).

Anyway one of the first problems compares two different juries.
The first is a one man jury where the juror gets it right with probability p. As there is only one juror his ruling applies.

The second is a three man jury with two jurors getting it right with probability p (independently) and the third coin-flipping juror who gets it right with probability 1/2 (again independently). In this jury majority rules. Which is to sy we need two of the three jurors to get things right.

Now the question asked is "which is better". A little algebra shows that they're both equally good, which is to say that the second jury reaches the right verdict with probability p.

So adding one "regular" (right with probability p) juror and one coin flipper doesn't change anything!  What if we ad yet another coin flipper and another regular juror? Now we need three of five correct. Now things do change.

A little more algebra shows that we now get the correct verdict with probability,

ppp+2.25ppq+0.75pqq=
pp(p+q)+1.25ppq+0.75pqq=
pp+0.75pq+0.5ppq=
p-0.25pq+0.5ppq=
p+pq(2p-1)/4

Which means that is p>1/2 that adding two jurors to our three (as we did to our one) now improves the verdict. Similarly if p<1/2 (why we're using jurors who're worse than coin flips I don't know but if we did) adding jurors 4 and 5 makes things worse.

Does anyone have any intuition for why adding the first pair of jurors does nothjign but adding the second pair helps?