At 7:34 is this really a paradox? Sure the possible side lengths won't have a different maximum or minimum value, but by describing the cubes according to their area or volume instead, I would think this would mean to imply that they vary evenly by the face area or volume, whichever it's saying. So naturally the distribution of possible side lengths won't be linear if we're assuming the other stated measure is mean to be linear. But really this paradox is also solved by just recognizing that we don't have enough information to find any probabilities yet until we're given some sort of definition of the distribution. Having now watched further to 8:04 to see the stated issue of "we get different results by reformulating the problem" I'm comfortable saying that this isn't actually the case. The example didn't reformulate the problem, it just asks three similar problems where they are all missing a key detail needed to answer them. So by leaving it out and assuming all possible values are equal, we end up asking different questions, not just the same question reworded. We could also express it in terms of x. Cubes of side length x where x a value from is 0 to 1. Cubes of volume x^3 where x a value from is 0 to 1. Like this, suddenly the "paradox" disappears completely.
@@TenriGenderless The problem with the classical view of probability is it says to assign a uniform distribution over the outcomes, but it’s unclear how what this uniform distribution should be “over.” When the cardinality of the set of outcomes is uncountably infinite, there are a lot of ways one can do this because you have to use measures and not just 1/n. As a result, the classical view alone doesn’t tell us how we should assign probability when the reference class is infinite. Are you arguing the problem isn’t unique to infinite sets? Or the problem comes from someplace different? Your point about not having enough information is well taken, and is actually a big criticism of the classical view - as Hajek points out, the classical view attempts to extract some information out of ignorance. You may find Jaynes’ principle of maximum entropy helpful, which tires to look at all possible statements of the problem and make a decision based on that (though of course Jaynes was a hardcore Bayesian). See also Alan Hajek - Interpretations of Probability in the Stanford Encyclopedia of Philosophy
No; the difference here is in the case you are mentioning, you are conditioning on the first child being a boy. The first child's sex is independent of the second, so P(2nd child is a boy | 1st child is a boy) = P(child is a boy) = 1/2. On the other hand, when you are told one of the two children is a boy, you don't know whether its the first or the second child. P(two boys | 1 of the two children is a boy) is 1/3. This is because there are two ways you can have one child being a boy and the other being a girl, (either the first is a boy and the second is a girl, or the first is a girl and the second is a boy) and only one way both are boys, hence the 1/3
@pogmumu3765 yeah; not counting the order was my initial reaction as well. Part of the confusion here is just that we typically ignore order, and part of this is due to the wording (one mathematician, LN Huong, described this problem as posed by a “troll” haha). I’ll do a video on the Monty Hall problem which used to bother me a lot, kind of a similar scenario
There is a big probability this channel gonna blow up
At 7:34 is this really a paradox? Sure the possible side lengths won't have a different maximum or minimum value, but by describing the cubes according to their area or volume instead, I would think this would mean to imply that they vary evenly by the face area or volume, whichever it's saying.
So naturally the distribution of possible side lengths won't be linear if we're assuming the other stated measure is mean to be linear.
But really this paradox is also solved by just recognizing that we don't have enough information to find any probabilities yet until we're given some sort of definition of the distribution.
Having now watched further to 8:04 to see the stated issue of "we get different results by reformulating the problem" I'm comfortable saying that this isn't actually the case. The example didn't reformulate the problem, it just asks three similar problems where they are all missing a key detail needed to answer them. So by leaving it out and assuming all possible values are equal, we end up asking different questions, not just the same question reworded.
We could also express it in terms of x. Cubes of side length x where x a value from is 0 to 1. Cubes of volume x^3 where x a value from is 0 to 1. Like this, suddenly the "paradox" disappears completely.
@@TenriGenderless The problem with the classical view of probability is it says to assign a uniform distribution over the outcomes, but it’s unclear how what this uniform distribution should be “over.” When the cardinality of the set of outcomes is uncountably infinite, there are a lot of ways one can do this because you have to use measures and not just 1/n. As a result, the classical view alone doesn’t tell us how we should assign probability when the reference class is infinite. Are you arguing the problem isn’t unique to infinite sets? Or the problem comes from someplace different?
Your point about not having enough information is well taken, and is actually a big criticism of the classical view - as Hajek points out, the classical view attempts to extract some information out of ignorance.
You may find Jaynes’ principle of maximum entropy helpful, which tires to look at all possible statements of the problem and make a decision based on that (though of course Jaynes was a hardcore Bayesian). See also Alan Hajek - Interpretations of Probability in the Stanford Encyclopedia of Philosophy
Nice
awesome video! thank you ^_^
Glad you enjoyed it :)
So if a child is born it's chance of being a male is reduced if there is already a brother?
No; the difference here is in the case you are mentioning, you are conditioning on the first child being a boy. The first child's sex is independent of the second, so P(2nd child is a boy | 1st child is a boy) = P(child is a boy) = 1/2.
On the other hand, when you are told one of the two children is a boy, you don't know whether its the first or the second child. P(two boys | 1 of the two children is a boy) is 1/3. This is because there are two ways you can have one child being a boy and the other being a girl, (either the first is a boy and the second is a girl, or the first is a girl and the second is a boy) and only one way both are boys, hence the 1/3
@@SignalProcessingWithPaul Thanks for the response. My thought was that we shouldn't count the order of the sons but that makes sense.
@pogmumu3765 yeah; not counting the order was my initial reaction as well. Part of the confusion here is just that we typically ignore order, and part of this is due to the wording (one mathematician, LN Huong, described this problem as posed by a “troll” haha).
I’ll do a video on the Monty Hall problem which used to bother me a lot, kind of a similar scenario