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Paradoxes of Probability and Other Statistical Strangeness

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Re: Paradoxes of Probability and Other Statistical Strangeness

#61

Post by adverbly »

Chinjut wrote: post #34
Retric wrote: post #33

Earlier quoted context omitted.

There are implicit conventions involved in reading this Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits. It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.

Sure, and if the quibble was along the lines of "You never explicitly said boys and girls are 50-50 distributed! You never explicitly said elder and younger childrens' birth genders are independent! You never explicitly said birth-days-of-the-week are uniformly…", then that would be fair, if pedantic. But this "You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you th…

I also fell into the same ambiguity trap, and I think that the objection about explicit wording is a fair one to make.

"What proportion of two-children families with a boy born on Tuesday have girls?" seems completely clear to me. I would have answered that question relatively quickly.

But the original question had me very confused. I felt a strong desire to ask more about the situation. A great deal of my intuition wanted to say that "well there is nothing special about Tuesday... Any boy that he has is going to be born on some day of the week, and if whatever day of the week that son is born on is included as this line item in the question, then that line item is irrelevant."

I wouldn't have fallen into that same trap in the case of the "What proportion of two-children families..." version because the "Any boy that he has is going to be born on some day of the week" logic doesn't apply.

Tuesday seemed like it might have been arbitrary in the original question, where it seems explicit in your rephrased version.

Re: Paradoxes of Probability and Other Statistical Strangeness

#62

Post by oh_sigh »

waqf wrote: post #55
oh_sigh wrote: post #52

Earlier quoted context omitted.

Is it probability actually zero, or just infinitely close to zero?

If it's a random real then the probability is zero. If it's an arbitrarily close approximation to a random real then the probability is arbitrarily close to zero.

How can the sum of infinitely many zero probabilities be 1? I can understand how the sum of infinitely many infinitely close to zero values can be 1, but not infinitely many exactly zero values

Re: Paradoxes of Probability and Other Statistical Strangeness

#63

Post by Retra »

oh_sigh wrote: post #52
Houshalter wrote: post #20

Earlier quoted context omitted.

The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.

Is it probability actually zero, or just infinitely close to zero?

Those are the same thing. You're just calling them by different names.

Re: Paradoxes of Probability and Other Statistical Strangeness

#64

Post by colonelxc »

boreas wrote: post #59

For those who might be interested, and in a slightly different vein than the examples in the article, there's the "sleeping beauty" paradox: https://en.wikipedia.org/wiki/Sleeping_Beauty_problem Basically, an agent is put to sleep and told they will be woken up once or twice, depending on the results of a fair coin flip, without the ability to remember other awakenings. What probability does the agent assign to the e…

I mostly find it interesting in that people could think that the chance is 1/3 (and that it may even be obvious!). After reading the description I can understand what they are getting at, but I think the conditional probability is messed up.

Instead of P(Monday | Heads) = P(Monday | Tails) = P(Tuesday | Tails) it is really P(Monday | Heads&Awake) = P(Monday | Tails&Awake) = P(Tuesday| Tails&Awake) or something like that. But the interviewer isn't asking about that, they are asking for the probability of the coin. The 3 positions are only exhaustive given that you are awake to be interviewed about them, not exhaustive of possible states (it's missing P(Tuesday | Heads&Asleep)). Since you're always awakened at least once, I find the argument that being awake has 'given you information that it is not tuesday AND heads' is pretty weak. While true, both heads and tails expect to be awoken while it is not both tuesday AND heads.

Re: Paradoxes of Probability and Other Statistical Strangeness

#65

Post by BeetleB »

georgewsinger wrote: post #3

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

This is virtually an axiom for continuous distributions.

One of the axioms of probability is that if you have an event (i.e. a set), then the probability of a countable union of disjoint sets is the sum of the probability of each set (event) occurring.

Assume a uniform distribution between 0 and 1. Now consider point sets of the rationals (i.e. the number 0.5 is represented by a set with just 0.5 in it). Since the distribution is uniform, each set has the same probability (i.e. the likelihood of picking a random rational).

Now consider this question: What is the probability of picking any rational between 0 and 1? Well, that's just the sum of the probabilities over all rationals (because it is a countable sum of disjoint sets). If the probability of picking any particular rational was non-zero, this sum would be infinite, which violates the laws of probability.

Thus, by convention, it's just simpler to define it to be 0.

There's no magic here. These properties were picked merely to make analysis with measure theory clean. Don't try to ascribe any real world meaning to picking a point.

Re: Paradoxes of Probability and Other Statistical Strangeness

#66

Post by Retra »

oh_sigh wrote: post #62
waqf wrote: post #55

Earlier quoted context omitted.

If it's a random real then the probability is zero. If it's an arbitrarily close approximation to a random real then the probability is arbitrarily close to zero.

How can the sum of infinitely many zero probabilities be 1? I can understand how the sum of infinitely many infinitely close to zero values can be 1, but not infinitely many exactly zero values

There's no such thing as a "sum of infinitely many" anything. What we are talking about is the limit of an infinite series, which behaves nothing at all like a sum.

Re: Paradoxes of Probability and Other Statistical Strangeness

#67

Post by Houshalter »

waqf wrote: post #54
bjl wrote: post #50

Earlier quoted context omitted.

You can certainly pick a random real number from the unit interval.

Really? Go on, then, pick one and tell us what it is (or at least tell us what your procedure was).

One example in a finite space and time setting would be selecting a random point on the ground. Say by dropping a ball there or something. The exact coordinates it lands is a random real number. But the probability that it landed on those exact coordinates is exactly 0, hence a paradox.

Re: Paradoxes of Probability and Other Statistical Strangeness

#68

Post by tzs »

colonelxc wrote: post #64
boreas wrote: post #59

For those who might be interested, and in a slightly different vein than the examples in the article, there's the "sleeping beauty" paradox: https://en.wikipedia.org/wiki/Sleeping_Beauty_problem Basically, an agent is put to sleep and told they will be woken up once or twice, depending on the results of a fair coin flip, without the ability to remember other awakenings. What probability does the agent assign to the e…

I mostly find it interesting in that people could think that the chance is 1/3 (and that it may even be obvious!). After reading the description I can understand what they are getting at, but I think the conditional probability is messed up. Instead of P(Monday | Heads) = P(Monday | Tails) = P(Tuesday | Tails) it is really P(Monday | Heads&Awake) = P(Monday | Tails&Awake) = P(Tuesday| Tails&Awake) or something like t…

Here is how one might decide that 1/3 is obvious. Imagine that N people simultaneous undergo the experiment, for a very large N.

Half of them end up in the heads group. They wake up on Monday and are questioned. Then they sleep until Wednesday and are released.

The other half end up in the tails group, and so are questioned twice (Monday and Tuesday) then released on Wednesday.

Because we gain no information during the experiment, we can make our decision before the experiment.

Let's count. There will be 3N/2 interviews conducted. N/2 of the will be 'heads' interviews and N will be 'tails' interviews. So going in, we can see that when someone experiences the event 'being asked about the coin', 1/3 of the time the coin will be heads and 2/3 o the time it will be tails. Hence, our credence in the coin being heads should be 1/3.

Here is a counterargument. Imagine a slightly different experiment. The people are not asked what their credence in the coin being heads is. They are asked to guess if it is heads or tails. If they are right, the experiment continues and they are eventually released. If they are wrong, this is noted, and the experiment continues until Wednesday, and then they are killed and their home planet is destroyed.

As before, we gain no information during the experiment, and so can decide our answer beforehand. No matter what strategy one picks for making that decision, there is a 50/50 chance that one ends up with a destroyed planet. That indicates that our credence in heads should be 1/2.

Re: Paradoxes of Probability and Other Statistical Strangeness

#69

Post by daxfohl »

Houshalter wrote: post #22
daxfohl wrote: post #2

"Paradox" is a pretty strong term. The items presented are more in the category of common errors and counter-intuitiveness.

https://en.wikipedia.org/wiki/Veridical_paradox

Fine, but I was left feeling blah by the "paradoxes" presented.

Re: Paradoxes of Probability and Other Statistical Strangeness

#70

Post by [deleted] »

Houshalter wrote: post #67
waqf wrote: post #54

Earlier quoted context omitted.

Really? Go on, then, pick one and tell us what it is (or at least tell us what your procedure was).

One example in a finite space and time setting would be selecting a random point on the ground. Say by dropping a ball there or something. The exact coordinates it lands is a random real number. But the probability that it landed on those exact coordinates is exactly 0, hence a paradox.

[deleted]
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