Which boy did I count twice?

Edit:

BAny/Boy1Tu in the above quote should be Boy2Any/Boy1Tu.

You could re-label boy1 and boy2 to be cat and dog and it won't change the probabilities - that would be CatTu/DogAny.

No, read it again. It's confusing as all getout, which is why they make the mistake, but EACH child can be born on ANY day of the week. The boy on Tuesday is a red herring, he doesn't factor into the probability for what day the second child can be born on at all. The two boys are not the same boys, they are individuals and their probabilities are individual. Re-label them Boy1 and Boy2 to make it clearer:

Here is the breakdown for the Boy1Tu/Boy2Any option:

Boy1Tu/Boy2Monday Boy1Tu/Boy2Tuesday Boy1Tu/Boy2Wednesday Boy1Tu/Boy2Thursday Boy1Tu/Boy2Friday Boy1Tu/Boy2Saturday Boy1Tu/Boy2Sunday

Then the BAny/Boy1Tu option:

Boy2Monday/Boy1Tu Boy2Tuesday/Boy1Tu Boy2Wednesday/Boy1Tu Boy2Thursday/Boy1Tu Boy2Friday/Boy1Tu Boy2Saturday/Boy1Tu Boy2Sunday/Boy1Tu

Seven options for both. For some reason they claim either BTu/Tuesday isn't an option, or Tuesday/BTu isn't an option, but I see no reason for this. Each boy is an individual, and each boy has a 1/7 probability of being born on a given day. In attempting to avoid counting evidence twice you've skipped counting a piece of evidence at all! In the original statement, they never said one and ONLY one boy was born on Tuesday, just that one was born on Tuesday. That's where they screwed up - they've denied the second boy the option of being born on Tuesday for no good reason.

A key insight that should have triggered their intuition that their method was wrong was that they state that if you can find a trait rarer than being born on Tuesday, like say being born on the 27th of October, then you'll approach 50% probability. That is true because the actual probability is 50%.

Just so it's clear, since it didn't seem super clear to me from the other comments, the solution to the Tuesday Boy problem given in that article is a really clever way to get the answer wrong.

The problem is the way they use the Tuesday information to confuse themselves. For some reason not stated in the problem anywhere, they assume that both boys cannot be born on Tuesday. I see no justification for this, as there is no natural justification for this, not even if they were born on the exact same day and not just the same day of the week! Twins exist! Using their same bizarre reasoning but adding the extra day they took out I get the correct answer of 50% (14/28), instead of the close but incorrect answer of 48% (13/27).

Using proper Bayesian updating from the prior probabilities of two children (25% boys, 50% one each, 25% girls) given the information that you have one boy, regardless of when he was born, gets you a 50% chance they're both boys. Since knowing only one of the sexes doesn't give any extra information regarding the probability of having one child of each sex, all of the probability for both being girls gets shifted to both being boys.

I'm talking about probability estimates. The actual probability of what happened is 1, because it is what happened. However, we don't know what happened, that's why we make a probability estimate in the first place!

Forcing yourself to commit to only one of two possibilities in the real world (which is what all of these analogies are supposed to tie back to), when there are a lot of initially low probability possibilities that are initially ignored (and rightly so), seems incredibly foolish.

Also, your analogy doesn't fit brazil84's murder example. What evidence does the lottery win give that allows us to adjust our probability estimate for how the gun was fired? I'm not sure where you're going with that, at all.

The real probability of however the bullet was fired is 100%. All we've been talking about are our probability estimates based on the limited evidence we have. They are necessarily incomplete. If new evidence makes both of our hypotheses less likely, then it's probably smart to check and see if a third hypotheses is now feasible, where it wasn't before.

The probability of both, in that case, plummets, and you should start looking at other explanations. Like, say, that the victim was shot with a rifle at close range, which only leaves a bullet in the body 1% of the time (or whatever).

It might be true that, between two hypotheses one is now more likely to be true than the other, but the probability for both still dropped, and your confidence in your pet hypothesis should still drop right along with its probability of being correct.

So say you have hypothesis X at 60% confidence and hypotheses Y at 40% New evidence comes along that shifts your confidence of X down to 20%, and Y down to 35%. Y didn't just "win". Y is now even more likely to be wrong than it was before the new evidence came it. The only substantive difference is that now X is probably wrong too. If you notice, there's 45% probability there we haven't accounted for. If this is all bound up in a single hypothesis Z, then Z is the one that is the most likely to be correct.

Contradictory evidence shouldn't make you more confident in your hypothesis.

4-step is what preceded 2-step. I say preceded, but it's not like 4-step has gone anywhere. It's still the most common beat pattern for electronic music. It's just a steady beat in 4/4 time with a kick drum on each beat, so it just goes boom boom boom boom with each measure, and it's super easy to dance to.

Techno and house are pretty much exclusively 4-step.

2-step runs at the same/similar speed as 4-step, and is still in 4/4 time, but the drum beat is split up and made more erratic. You'll often have several drum rhythms going on simultaneously. The effect is that the beat sounds like it is sort of stuttering, sort of like this: boom boom pause boom pause pause boom boom pause boom boom boom (that's three measure's worth there). I think Garage was the only real 2-step going on before dubstep, but I'm not real clear on that part of it.

Dubstep gets it beat patterns from 2-step (thus the "step" in the name).

The "Dub" comes from the reggae tradition of sampling pop songs to build a record in an afternoon. That's why the vast majority of dubstep tracks are remixes - it's just how you make dubstep. The build ups and drops that are so popular are not necessary for dubstep, and just because a song has that stuff doesn't make it dubstep. They are just a natural fit for DJ's (who like to control the energy of a crowd) and a 2-step beat pattern.

It's not really dubstep if it isn't heavily sampled with an erratic 4/4 beat (aka 2-step).

What else could it be?

The break distance bias found in the papers?

You can't use two pieces of contradictory evidence to support the same argument. If the most highly contested cases still have a chance at success, finding 0% success rate at the furthest distance from the last break (because they are the longest cases and therefore placed last) should not increase your belief that there is no bias at work. It should reduce it. How significantly your belief is reduced depends on just how likely you would see 0% success rates at a high distance from break due only to scheduling, but I can't see any way it could legitimately raise your belief that there is no bias.

I agree that any contested case should be longer than an uncontested, however are there not cases where the prosecution simply doesn't need to go through a lengthy argument to prove their case? Prosecution lays out X, Y, and Z evidence that is definitive, and therefore the prosecution doesn't need to spend a lot of time arguing. Are these types of cases not generally shorter than cases that are contested but more likely to succeed? Or does a lengthy defense attempting to weasel out of the evidence make up for a short prosecution? And are these specific cases few and far between?

I doubt it, but there chances of success are surely worse than uncontested applications.

In this context I could believe a very low success rate, but the researchers found a 0% success rate for a number of courts. That still makes me suspicious. I'm still not sure what "very low success rate" means for parole hearings though. Is 20% low? Is it more like 5%? Somewhere in between? Obviously, the lower a reasonable success rate for these types of cases the more likely you'll see 0% rates in different courts, just based on chance.

I don't know enough about the details of Israeli parole hearings to speak definitively about that. I can say >as a lawyer that many times I have made uncontested applications which were denied. This was not in a >criminal or parole context.

Fair point. Like I said above I'm not really sure what my expectations should be for a reasonable success rate in these types of cases (or cases in general). Question though, did these applications tend to be closer to or further from a break than your more successful uncontested applications? (obviously purely anecdotal, but I'm sure you see my point)

I can certainly buy that, but would there really be zero people who apply even though they don't have much chance of winning? I know a few stubborn people who I would expect to apply anyway even if they didn't have much chance of success. I'd be surprised to find out that the prison system has an insignificant number of people who are like that as well.

Also, do the most highly contested applications (and therefore the longest, and therefore placed last on the docket) really have 0% chance of success? If so, would not those applications be better off not applying at all? It seems to contradict the idea that an application with 0% chance of success would not be filed.

Lastly, with only a 65% approval rate for the early applications, I'm pretty surprised that the prosecution doesn't care about them very much. If they were completely uncontested wouldn't you expect closer to a 100% success rate?

I don't see any reason there wouldn't be the inverse as well. That is, applications which are immediately rejected, and therefore quite short.

I also find it suspicious that the most highly contested applications would also be the least likely to be approved. Presumably these are the ones which are borderline, and require much argument, pro and con, to come to a decision. Immediate rejections wouldn't require long arguments, and neither would immediate acceptances. Under the above hypothesis, both of these types of cases should be early in the session.

If there were no bias and the cases were arranged by length, I would expect to see nearly 50% of the lengthy contested cases, to be accepted. Or, if that many are simply not accepted, some number significantly greater than 0%. For a slightly different arrangement, if the immediate rejections were placed last in the session I would expect the acceptance rates to start at nearly 100% and progress down to 0%. Unless, of course, there is no such thing as a quick, immediate acceptance parole case, in which case the argument doesn't work anyway. If they are all long there isn't much point in arranging by likelihood to be accepted.