A logic question regarding mathematics education

[psionl0]

There is no such thing as getting bogged down in semantics in mathematics.

You are proof that there is.

The definition of probability is straightforward enough. However, by getting bogged down with objections over terms such as "inherent property", "assigned probabilities", or "ascertained properties" and insisting that there is no such thing as "experimental probability" (because your ancient text book didn't mention it) you have obfuscated probability out of existence.

If you had your way, nobody would be able to come up with satisfactory answer to 1+1=?

 

[jt512]

You are proof that there is.

The definition of probability is straightforward enough. However, by getting bogged down with objections over terms such as "inherent property", "assigned probabilities", or "ascertained properties" and insisting that there is no such thing as "experimental probability" (because your ancient text book didn't mention it) you have obfuscated probability out of existence.

If you had your way, nobody would be able to come up with satisfactory answer to 1+1=?

Your post is idiotic, but to play your game for a moment, 1+1 would have a separate theoretical sum and an experimental sum.

 

[dakotajudo]

It makes no theoretical sense, and so it shouldn't be there, or anywhere else. The problem is the word "assigned." It treats probability as something we give to the coin, rather than a property of the coin that we are attempting to infer.

In classical statistics the probability of an event, say that event that a tossed coin lands heads, is the long-term relative frequency with which the event occurs. Roughly speaking, if we flipped the coin literally infinitely many times, the probability would be the proportion of times that the coin landed heads. However, we can ever only flip the coin a finite number of times, so the true probability of it landing heads is unobservable. This leaves us with two alternatives: (1) we can attempt to deduce the probability from the physical properties of the coin, or (2) we can estimate the probability by carrying out an experiment.* But in neither of these cases are we "assigning" to the coin its probability. That probability is an inherent property of the coin itself. Saying that we assign probabilities to the coin obfuscates the fact that we are attempting to ascertain a property of the coin.

*A third method, Bayesian statistics, would combine (1) and (2).

If you're using measure-theoretic probability, then "assignment" is a valid term. Assigning a probability to an event is equivalent to measuring the likelihood of that event. More specifically, a probability measure function mapping a set of events to a real number value from 0 to 1.

In classical statistics, probability is not a property of the coin - it is a property of the random experiment involving the coin, because, of course, the action of tossing the coin is not inherent in the coin, but it is part of the experiment. It can be argued that under certain experimental conditions, it is physically impossible for a coin to be biased, and we can assign the theoretical value for a fair coin - that is, probability of heads=0.5 (see http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf).

 

[jt512]

In classical statistics, probability is not a property of the coin - it is a property of the random experiment involving the coin...

Now that I would call getting bogged down in semantics.

 

[psionl0]

Now that I would call getting bogged down in semantics.

LOL you're not very good at this. I could ask what your probability is. (After all, it is a property of you, isn't it?)

 

[jt512]

LOL you're not very good at this. I could ask what your probability is. (After all, it is a property of you, isn't it?)

Sure, in Bayesian statistics. But not in frequentist statistics.

 

[psionl0]

Sure, in Bayesian statistics. But not in frequentist statistics.

Now that I would call getting bogged down in semantics.

 

[jt512]

Now that I would call getting bogged down in semantics.

Then you wouldn't know what you are talking about.

 

[phiwum]

I'm teaching a high school-level statistics course. There is no class set of textbooks but I have a teachers edition I'm loosely using as a curriculum guide.

I was browsing through "measures of dispersal" and found this sentence:



I asked students if they could spot a logical flaw in the sentence. They couldn't immediately see it. That "because" bothered me. It seems both clauses are saying the same thing: Because the mean deviation is smaller, the mean deviation is smaller. It's not cause and effect; the book is just stating the same thing 2 different ways. "Begging the question" IMO.

Am I reading that sentence correctly, and is my analysis valid?

I'd like to challenges students with more such logic questions but it's really not the focus of the course. I have quite a bit of leeway though.

Minoosh,

You're right that the "because " is not about cause and effect nor about explanation generally. It is a premise indicator. Think of this as a simple argument. The clause following "because" is a premise and the rest a conclusion.

It's a little odd to say, in normal discussions, Px because Qx when P and Q are equivalent by definition, but there's no fallacy in doing so.

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