I've been asked to set a competition style maths contest for first year students at wits. For some reason or another the organizers decided to call this a (the?) math bee. The goal is 20 multiple choice questions of a level somewhere between the SAMO Seniors 2nd round and the SATMO.

I'd like to avoid the paper having too many of my own personal biases. To that end this post is a call for questions. If anyone has any that seem appropriate then please e-mail me them to me, I will thank you via blog post.

## Sunday, May 24, 2015

## 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

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?

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?

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