My favorite statistical/probability paradox has always been the birthday paradox.
I don't know if Monty hall problem counts as a paradox, but that is quite high on my favourite list of counterintuitive probability results.
Paradoxes of Probability and Other Statistical Strangeness
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Re: Paradoxes of Probability and Other Statistical Strangeness
#22"Paradox" is a pretty strong term. The items presented are more in the category of common errors and counter-intuitiveness.
Re: Paradoxes of Probability and Other Statistical Strangeness
#23Earlier quoted context omitted.
Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.
Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense. In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained b…
> In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer
An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.
Re: Paradoxes of Probability and Other Statistical Strangeness
#24By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.
The comments to this video actually say (with proof) that this was an error in the video.
Re: Paradoxes of Probability and Other Statistical Strangeness
#25By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.
The comments to this video actually say (with proof) that this was an error in the video.
There seems to be quite a bit of debate about it in the comments and I'm not sure who to believe. At one point someone coded a simulation to test it and the results were as predicted by the video. Even if the video is incorrect, the fact it's so confusing still makes it an interesting paradox.
Re: Paradoxes of Probability and Other Statistical Strangeness
#26By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.
It's fairly simple.
If you flip 2 coins then say whatever the first coin was the the odds if you said H was HH, or HT and if you said T it would be TH, TT. However, if you flip two coins and then say if you got at least one head independently from whatever you flipped then the odds you have 3 options HT, HH, TH with equal odds.
So, the question is if the full statement was based on the data or only the truth value of the statement is based on the data.
PS: Now assuming it's truth value is based on data. if you look at all options there are 14 gender day combinations per kid and 14 * 14 = 196 gender day combinations in totoal. Only 14 of of those 196 start BT which is then split evenly 7 BTB_, 7 BTG_. However that leaves 196 - 14 other options to consider. 7 * 14 of them Start G, and 6 * 14 of them start with B not on a Tuesday, but out of those you only keep 1/14 as you need BT on the second roll. Now add them up 13B and 14G out of (13 + 14) = 27. Or 13/27 B, and 14/27G.
Re: Paradoxes of Probability and Other Statistical Strangeness
#27By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.
Your problem is that you are thinking there's a "the boy". But there's not a "the boy". Mr. Jones could have two boys. He could have two boys both born on Tuesday, even. The term "the boy" does not denote any particular boy, in that case, and causes you to think about the situation erroneously.
If the question were "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out of those cases where Kid 1 is a boy born on Tuesday, what proportion are cases where Kid 2 is a girl?", then the answer would indeed be a straightforward 50%; the status of Kid 1 is entirely independent of the status of Kid 2.
But that's not the question. The question is "There's Kid 1 and Kid 2, each independently selected with random gender and birth-day-of-the-week. Out of those cases where at least one (either one, and possibly both) of Kid 1 and Kid 2 is a boy born on Tuesday, what proportion are cases where at least one of Kid 1 and Kid 2 is a girl?".
This is very different, and of course just drawing out the possibilities (all 2 * 7 * 2 * 7 equiprobable-by-stipulation choices of gender and birth-day-of-the-week for Kid 1 and Kid 2) and circling which pairs of subsets are the relevant ones for the two questions reveals the difference, the probabilities for either question elementarily calculable in this way by basic counting.
Re: Paradoxes of Probability and Other Statistical Strangeness
#28By far the most unintuitive paradox for me personally is the one presented here: https://youtu.be/go3xtDdsNQM?t=3m27s "Mr. Jones has 2 children. What is the probability he has a girl if he has a boy born on Tuesday?" Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre.
The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this.
Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3.
Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then the probability that he has a girl is less than 2/3, because having two boys gives (about) double the chance for one of them to have been born on a Tuesday.
However, suppose that you asked Mr. Jones weather he has a boy, and if so what day his eldest boy was born on, and he says "yes, and on Tuesday". Then the probability that he also has a girl is again exactly 2/3.
Wikipedia has a detailed explanation: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
Re: Paradoxes of Probability and Other Statistical Strangeness
#29Earlier quoted context omitted.
I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.
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.
Non-countable sets defy intuition on several ways. The silver lining is that we don't have any evidence a non-countable thing exist on the real world.
I don't think anybody even has a procedure for gathering that kind of evidence.
Re: Paradoxes of Probability and Other Statistical Strangeness
#30Earlier quoted context omitted.
I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.
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.
I would argue that the distribution you used to pick a random number was not uniform. Not all the real number were equaliy likely to be picked by you. Hence the probability for some numbers was > 0.