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## Question

You're suing for damages and have a slam dunk case on the evidence.

So you want a jury that will get it right. The jury pool comprises Thinkers

and Gamblers. All Thinkers have a probability p of getting the right answer.

All Gamblers flip a fair coin.

You have a choice of juries:

1. Two Thinkers and one Gambler. You need two votes out of the three.
2. One Thinker. You need his vote.

Which jury do you pick?

This is a deceptively easy problem, and you should be able to do it in your head.

Have fun.

## Recommended Posts

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Given: no faulty evidence, no paid off jury members, and no contacts higher in the system. Either solution is the key. Solution one: The thinkers both see the evidence, therefore, they will both vote in your favor despite the gamblers vote. Solution two: The thinker will see you are correct and give you his vote. One catch to the problem that nullifies all of this? I believe juries are slightly larger in size than one or three people.

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Given: no faulty evidence, no paid off jury members, and no contacts higher in the system. Either solution is the key. Solution one: The thinkers both see the evidence, therefore, they will both vote in your favor despite the gamblers vote. Solution two: The thinker will see you are correct and give you his vote. One catch to the problem that nullifies all of this? I believe juries are slightly larger in size than one or three people.

Be careful not to assume a value for p - only that it's in [0, 1].

That is, the Thinkers might be bad thinkers.

This is a special trial, so the jury can be any size.

If you like, let it be the choice of a three-person jury or no jury at all and the judge decides.

The judge, by the way is a Thinker and also has a probability p of getting it right.

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2

1/2 X p, or p X p will always be less than (or equal to) p.

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...that it doesn't matter which jury you pick. If all Thinkers are the same then in the case of the first jury both Thinkers will vote the same way and will determine the outcome regardless of the Gambler's vote. In the second case one Thinker determines the outcome. So in both cases you need to convince the Thinker(s) with your evidence to get them to vote in your favor.

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the one thinker. p*p and p*.5 will always be less than or equal to p.

Edit: The fact that all Thinkers have the same probability of making the right choice doesn't mean they will always choose the same way as each other.

Edited by benjer3
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option 1. is twice as likely?

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Maybe I misinterpreted the OP in thinking that all the Thinkers will come to the same conclusion, but nonetheless the result is the same. The choice doesn't matter. The probability of the favorable outcome is p for both juries. Here is why...

The second case is trivial and the probability of the favorable outcome is p.

The probability of the favorable outcome with the first jury is the sum of probabilities of possible favorable outcomes. There are 4 favorable outcomes. Two favorable outcomes when both Thinkers vote in your favor have the probability of p2/2 each. The other two when one Thinker votes against you have the probability of p(1-p)/2. The sum of these probabilities is p.

Here is the complete table of outcomes and their probabilities:

```
T1(p)	T2(p)	G(1/2)	Probability	Favorable

Y	Y	Y	p*p*1/2	        Yes

Y	Y	N	p*p*1/2	        Yes

Y	N	Y	p*(1-p)*1/2	Yes

Y	N	N	p*(1-p)*1/2	No

N	Y	Y	p*(1-p)*1/2	Yes

N	Y	N	p*(1-p)*1/2	No

N	N	Y	(1-p)*(1-p)*1/2	No

N	N	N	(1-p)*(1-p)*1/2	No

```

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although the indviual chances of getting two thikers to vote for you or a thinker and a gambler to vote for you are smaller than a thinker alone, if you combine probabilites...

p*p*1/2 +p*p*1/2 +p*(1-p)*1/2 +(1-p)*p*1/2 = p^2 +p -p^2 = p.

(all 3) (2 think) (1 think 1 gam) (1 think 1 gam)

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I think the 1st choice would be better.

The chance of both thinkers voting for you is p2

The chance of a thinker and a gambler voting for you is p*1/2*2 which equals p. (there are 2 thinkers)

So the chance of the jury voting in your favor would be p2+p which is more than the 2nd choice(p).

Although, I didn't include the probability of someone not getting it right, I didn't include the chance of them all getting it right either.

Edited by t8t8t8
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agree with the "same" votes. didnt subtract the times when the gambler voted wrong when figuring the probability of 1-neither thinker getting the correct verdict. another quickie way to look at it is to say p = .5. odds of getting two or more heads when flipping three coins ==> .5

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option one. you guys keep saying it as if thinkers are on your side and are always right. do not forget that the Gambler has a 1/2 chance of believing you.

And if you have a slam dunk case, it doesnt matter.

Edited by Aaryan
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option one. you guys keep saying it as if thinkers are on your side and are always right. do not forget that the Gambler has a 1/2 chance of believing you.

And if you have a slam dunk case, it doesnt matter.

Do the initials OJ ring a bell?

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It is not going to matter which you pick you have the same chance.

Since we do not have p the thinkers could be anywhere between 0% likely to correctly determine your case and 100% likely. On average this makes them 50% likely to determine it correctly (you might as well have a gambler instead).

For three gamblers with a fair coin the various ways the coin could fall are

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

There are 4 heads and 4 tails giving you a 50% chance of winning the case. Which would be exactly the same chances as having one juror

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p x p + 1/2 x p + 1/2 x p = p + p^2

only if p > 0.37

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Do the initials OJ ring a bell?

True, but if the thinkers are fair, it does not matter. my point was, some of the people were forgetting that Gamblers had a 1/2 chance of getting it right.

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Do the initials OJ ring a bell?

True, but if the thinkers are fair, it does not matter. my point was, some of the people were forgetting that Gamblers had a 1/2 chance of getting it right.

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