Class 11: One Son Born Tuesday & Relevance
Uncertainty & Probability Theory: The Logic of Science
Uncertainty & Probability Theory: The Logic of Science
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HOMEWORK: RELEVANCE. A psychic is guessing cards from a standard poker deck of 52 cards. She must guess the exact card, suit & value. When she guesses right, she receives a chocolate. When she is wrong, she gets nothing. In the first 10 cards, she got 4 right and 6 wrong. What is the probability P = "She gets the 11th correct"? The exact number is good if you can get it, but it's much, much more important to explain your reasoning. I mean, what is your Q?
Lecture
Tuesday’s Child
Last week’s homework was this: You meet a guy who says, “I have two children, one of whom is a boy who was born on a Tuesday.” What’s the probability, given this and only this Q, that both this guy’s children are boys?
Let me not answer that right away. Instead, look at this, which is everything that can happen with two kids, recalling that part of our Q is always that there are two kids:
First Boy, Second Girl
First Boy, Second Boy
First Girl, Second Boy
First Girl, Second Girl
Given only that, and not some host of tacit premises about human births, what’s the probability of both boys? Obviously (well, we haven’t yet justified other numbers for probability, which we do next week, but pretend we have), it’s 1/4. Because there’s only 1 out of 4 ways to get both boys.
Now suppose you add to Q that the first child is a boy. What is the new probability? Right: it’s 1/2. Because we only consider those times where the first child was a boy, which can only happen two ways, one of which results in a second boy.
Understand, it would not matter if these were children or doubloons, a B one and a G one. Or whatever. This question assumes nothing about biology. It only mentions certain states—just two of them as possibilities. That’s it. You must always be wary of tacit and implicit premises!
Now suppose instead of assuming the first child is a boy, you assume the first or the second can be a boy. We must now consider only those instances in which the first or second child was a boy. This is three possibilities, only one of which resulted in two boys. That gives a probability of 1/3.
So we have three different probabilities for two boys, depending on which information we condition on.
Which brings up the most magnificent point: probability says nothing about cause. Do not think about birth order causing anything. There is no cause. The timing is only relevant because it opens up possibilities. In the last Q, we open more possibilities for girls. That’s it. Nothing more.
Let’s expand the example. Guy says one of his two children is a boy born on the weekend.
Now this is where the big trouble starts. Weekend is irrelevant in any causal sense to what happens to the other birth. Or even the first birth. But we naturally think of cause. We can see there is no causal relevance, so we give a probability with that Q in mind. But not this Q.
The big step is to supply the tacit premise “Well, if one is born on a weekend, the other might be born on a weekday.” This is now two possibilities. If we supply that premise, then we get a whole different probability. Here are the possibilities. To save writing, the order is birth order, B and G, and Y is for weekday and D is for weekend.
BY, BY
BY, BD
BY, GY
BY, GD
BD, BY
BD, BD
BD, GY
BD, GD
GY, BY
GY, BD
GY, GY
GY, GD
GD, BY
GD, BD
GD, GY
GD, GD
If I next ask what’s the chance for two boys, given all this, and only all this, we have 4 out of 16 possibilities, for a probability of 1/4.
If I ask, what’s the chance for two boys given the first was born on a weekday (Y), then we count only those tines in which a first boy was born on a weekday. That’s 4 possibilities 2 of which were both boys for a probability of 1/2. If I ask, what’s the chance for two boys given the first was a boy, that’s 8 possibilities out of which 4 are both boys, again a probability of 1/2.
But if I ask, what’s the chance for two boys given either the first or the second was born on a weekday, then we have 7 possibilities, only 3 of which result in two boys. For a probability of 3/7!
The boy born Tuesday is exactly like that, but with implied (tacit) premises of the other seven days of the week being possibilities.
It turns out we can generalize this probability to a formula. I haven’t seen anybody do this before. But it turns out to be:
(2n – 1) / (4n – 1),
where n is the number of possibilities we grant for birth situations. When only the order mattered, n = 1. In the weekend/weekday example, we have n = 2. In the Tuesday example, we have n = 7.
That gives probabilities:
n | Pr = (2n – 1) / (4n – 1)
————————————
1, 1/3
2, 3/7
7, 13/27
And so on. Obviously, in the limit, the probability goes to 1/2, as our intuition wanted it to be. In the limit, there are infinite possibilities for birth possibilities, which then turns out to be irrelevant to our calculation.
As to relevance this excerpt.
Like with Monty Hall, if you doubt all this, you can play the game yourself and see that the probabilities are correct. Do just the birth order because that’s easiest.
This is an excerpt from Chapter 4 of Uncertainty.All the references have been removed.
We need the idea of relevant and irrelevant evidence. Irrelevance occurs when the probability of some proposition, given whatever evidence we already have, is not changed in the face of different and new evidence. The probability of P would obviously not change it we merely repeated evidence in Q which was deducible from extant evidence. That is, the probability of P given Q is identical to the probability of P given “Q & Q”; and if Q logically implies W, then the probability of P given “Q & W” must equal the probability of P given Q—but it is not necessarily the case that $Pr(P|Q)=Pr(P|W)$.
Causality and its lack also plays a role. For instance, the proposition “This cargo ship is over 100,000 tons” has high probability given the evidence, “Most cargo ships are over 100,000 tons, and this ship is a cargo ship.” If I add the true premise “Einstein enjoyed tobacco” the probability doesn’t change: the evidence is irrelevant because we cannot identify any causal connection, no matter how complex, between the propositions. This supposition falls short of proof, naturally, but that’s because we are in the realm of the contingent. I cannot prove to you that Einstein’s hobby is unrelated to cruise ship weights; nevertheless, induction demands it. There is no identifiable logical connection between cargo ships and Einstein’s smoking. Induction lurks here as everywhere. The reason we know Einstein’s smoking has no bearing on the question is because of induction.
The opposite of irrelevant is relevant. If I add the premise “This cargo ship is smaller than most”, the probability of our proposition changes: by how much is not known, of course, but it is obvious, given our (tacit) understanding of the English language, that this new evidence is probative, that it is relevant, and it is relevant because it is determinative.
It makes a difference when evidence is introduced. If P = “John was the killer” and we began with Q_1 = “The murderer was a dentist and John is a dentist” the additional minor premise Q_2 = “John graduated from Honest Ben’s Dental College” is irrelevant because we already know John is dentist. But if all we had was Q_3 = “The murder was a dentist”, adding Q_2 (and leaving out Q_1) is relevant. This point is taken up again in models (especially time series). Relevance is itself conditional on the accepted premises.
Evidential importance or weight to a fixed proposition is measured, when it can be measured, but how much the probability of a proposition changes on the addition of new evidence. Importance is thus also relative to the information already present in the premises. If we start with Q = “At least 6 of the 11 balls are orange, and this is a ball”, the probability of P = “This ball is orange” can be calculated (see below). If we add to Q the evidence “There are 6 orange balls”, the probability of P conditional on the augmented Q is 6/11. How much has the probability “moved”? Clearly by some appreciable and happy amount. We have reduced the choices from six to one. Weight will not always be quantitative, just like probability is not always quantitative. For instance, we could start with Q = “Some of the 11 balls are orange, and this is a ball”. As before, we revisit this technique when we discuss modelling.
How relevancy, irrelevancy, importance, and weight are measured, using for instance the techniques of entropy and the like are, of course, of tremendous practical interest. But those techniques, while interesting, are incidental to the philosophical points made here. Jaynes is an ideal reference work.
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Sorry, all. I goofed and left the HOMEWORK out of the original. It's now added. But those who got emails won't see it.
The simplicity of your explanations just drive home the stupidity involved in the massive wastes of time and money involved in the entirety of the "Climate" kerfuffle. I understand that a some of the people driving it are doing so because of the tsunami of government money being poured into it, or because of the power it gives them, but I absolutely cannot understand how any rational person can buy into the hysteria surrounding anthropogenic climate change given the 100% failure rate of the predictions to this point, and given the obviousness of the vast numbers of vague assumptions and bad data poured into all the models used as justification.
I just don't understand how we have gotten to the point where the government can use "climate change" as an excuse to dictate to me what kind of light bulb I'm allowed to have, what kind of dishwasher I'm allowed to buy, and to make the building codes restrictive to the point of making the building of a home ruinously expensive, and so many people simply nod their heads without the slightest moment of doubt.
There are so many things wrong with all of it, and it's all built on such purely speculative assumptions. I just don't get it.