We started the week with a few words about estate taxes.

Then it was on to the justice system. On Tuesday I offered a little quiz on recognizing reasonable doubt (or its absence) in an exceptionally simple environment. Some readers thought that environment was too simple to be interesting. On the contrary, it’s simplicity is what **makes** it so interesting. If we can’t recognize reasonable doubt in such a simple environment, how can we ever recognize it in the courtroom?

On Wednesday, we talked about the appropriate numerical cutoff for reasonable doubt, and on Thursday we took a step back and asked what principles we should apply in choosing that cutoff. On Friday, I decried the dereliction of duty by judges and legislators who refuse to tell us what cutoff they have in mind when they use the word “reasonable”.

Some commenters thought that giving jurors a precise numerical standard was asking them to think more “mathematically” (whatever that means) than we can reasonably expect. But there’s no mathematics involved in telling a juror that he should convict if he believes that in 100 similar cases, at least 93 of the defendants will be guilty. No mathematics, that is, beyond the ability to count to 100, which is, I think, something we already expect of our jurors.

If I don’t tell you what “reasonable” means, then “beyond a reasonable doubt” makes as much sense as “beyond a gribzle doubt”. Judges could, if they wanted to, tell juries to convict if the evidence convinces them beyond a gribzle doubt, and then refuse to reveal what “gribzle” means. I don’t see how that system would differ substantially from the one we have now.

lewis carrol would make a great judge.

‘whiffers and specklefrotz of the jury, do you understand your instructions?’

spot on, doc. ‘I strongly suspect it’s primarily to make the jury’s job as hard as possible so as to drag out the deliberations and make more work for judges.’

also interesting is that we actually vote on whether or not some of them keep their jobs. i just randomly put yes’s and no’s down in an attempt to offset my complete lack of knowledge about their job performance.

Can it be simply that “reasonable doubt” has no fixed definition, but that it varies from time to time, that we belive the standard applicable varies from crime to crime, and that we expect the jury (as a kind of sample of society) to act as a proxy for society in taking that decision? Hayek, distributed knowledge, and all that?

No, you are asking jurors to think much more mathematically than a count to 100. You are asking them to conceptualize a set of hypothetical with similar evidence, and to make corresponding judgments.

Since people here really seem to be into statistics, I have a question. Today at a coffeehouse where they write interesting facts up on the board, it said “If you take a random group of 23 people, odds are that 2 of them will have the same birthday.” I started thinking about what the odds actually would be, and the numbers I’ve been getting wouldn’t merit that statement. However, since you guys always come up with answers that I would never have thought of, I was thinking maybe someone would be able to provide some insight on this. Thanks for any help!

Scam: If there are three people, what do you get for the probability that they all have different birthdays?

99.7%

OK. I think I’ve got it down a little better now. With some help from a friend who’s much smarter than myself, I’m now getting about 50.7% for the 23 people. It definitely goes against intuition being that high, but I guess it is. I wish we could do fun stuff like this in stats class.

Scam,

To be precise,you’ve computed the probability that two OR MORE people have the same birthday. This will get closer to one if you add more people to the room.

For more fun, you can try to determine the probability that EXACTLY two have the same birthday. Then you can figure out how many people you need in the room to make this as large as possible.

Well I’m having fun with numbers. Although my calculator not being able to do factorials of more than 100 isn’t helping much. I found out that with 23 people you can make 253 different combinations, which helps frame the question to make it more intuitive. Also, probably not coincidentally, it would take 253 people to make the odds greater than 50% that someone else has the same birthday as yourself.

As far as figuring out the probability that exactly two have the same birthday, I couldn’t really figure out any math behind it. However, after you get to 4 people the percentage will start to be smaller than the “at least” percentage. Also, unlike the “at least” percentage which will eventually grow to 100%, the “exactly” percent will start to diminish at some point on the graph making it look like a parabola. So I guess the question is to find the max on that graph. Any suggestions on how to go about doing that?