5 ms·
As the currently top voted comment does not get it, I try to intuitively explain the paradox. No, it is not ambiguity of language. It says formally: I have tw
by nadam 15y ago
As the currently top voted comment does not get it, I try to intuitively explain the paradox.
No, it is not ambiguity of language. It says formally:
I have two children. There exists a child of mine who is (boy and born on tuesday).
And yes, the probability of the other child being a boy is 13/27.
To understand this, try a more extreme case:
When a child is born we generate a random number: rnd(1billion)
Now the man says:
'I have two children. There exists one of them which is boy with generated id=456765234'
As we can see, the probability of the other child being a boy approaches 1/2 as we increase the bound of our random number!
Why is this intuitively?
Because by using a very specific id we are closer and closer to specifying one of the boys, as the probability of the id not being unique decreases. We are closer and closer to saying this:
'Randomly pick one of my children. He is a boy. What is the probability of the other being a boy?'
Yes, that would be 1/2.
Edit.:
Seriously downvoting this??? For most commenters the correct result was not intuitive. (It was not intuitive for me at first also). I tried to make this intuitive to them. (a bit deeper than usual paradxes though) Or do you still think the 13/27 result is not correct? In that case you are fighting mathematical truth with the downvote button.
Edit: Thanks, corrected probability from 2 to 1/2
- demallien 15y agoNo, it is not ambiguity of language. This is a big call to make, seeing as English is not your native language. As a native English speaker, I find it very ambiguous as to which set of probabilities I should be counting.
- nadam 15y agoI meant that the paradoxical nature is not resolved by simply formalizing the statement into first order logic the following way: "I have 2 children. Exists child where (Born_on_tuesday(child) and Boy(child))." (Some other paradoxes completely disappear when you try to write the statement into first order logic. It is not the case here.) This is what I wanted to say.
- demallien 15y agoYou see, the problem is that even your latest attempt to describe the question is ambiguous - the solution depends on how 'I' was selected. And if ever you describe that in a non-ambiguous manner, I suspect that the answer would no longer seem paradoxial. In other words, ambiguity in language is at the very heart of this problem.
- ars 15y agoWhy can't the other boy also be born on Tuesday?
- nadam 15y agoThe other can also be born on Tuesday. (generated id = 1 in rnd(7)) And the other can also have a generated id = 456783123 in rnd(1billion). But as the bound of the rnd is bigger and bigger, the probability of that approaches to zero. So as we know more and more information of one of the children, we are closer and closer to simply uniquely identifying him, so the probability of the other being a boy approaches 2.
- ars 15y agoIf both can be born on Tuesday then telling me one was born on Tuesday does not uniquely identify him! It's not unique if both can do it. And since it's not unique the rest of your analysis is based on a faulty assumption. (BTW I did not downmod you, in case you were wondering.)
- nadam 15y agoIt is about specifying a child, and then saying it is a boy. The one extreme is: one of my children is a boy. in this case the other is a boy with 1/3 probability. The other extreme is: One of my children has a national unique id=... He is a boy. The other is a boy with 1/2 probability. And other cases are in between. The more information you provide on the first child (the less chance there is that the other can have the same property, the more close you are to 1/2. Of course in the rnd(7) example we are in-between: 1/3 < 13/27 < 1/2
- ars 15y ago"the less chance there is that the other can have the same property" No! That is not true! The two events are independent, specifying information on one has zero impact on the other.
- tallanvor 15y agoWell, you're talking about probabilities being greater than 1... That doesn't really make sense in this discussion.
- dmd 15y agoI think it's fairly clear that nadam means 1/2, not 2.
- user24 15y agoit is now he's edited it. Wasn't clear before, to me at least. But then I "don't get it" so what do I know ;) edit: can't reply to you as HN isn't giving me the option. But: No worries, I didn't take it personally, just a little dig :)
- copper 15y agoPersonally, I find Tanya Khovanova's explanations (also linked to by the article) much easier to read: she's discussed this and related problems quite a few times. http://blog.tanyakhovanova.com/?p=221 http://blog.tanyakhovanova.com/?p=221
- nadam 15y agoI did not mean 'not getting it' in a negative way. (English is not my mother language.) Getting a paradox or not is a state. First I did not get it also. And I only wrote my comment, because at the time it was almost a consensus in this thread that the paradox is an uninteresting language ambiguity. I wanted to say that it is more interesting than that.
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- zerostar07 15y agoRandomly pick one of my children. He is a boy. What is the probability of the other being a boy? Yes, that would be 1/2. To be clear, that probability would be 1/3. But I believe i understand where you are getting at. You have to phrase the question in the form "pick one of my 2 children, he is a boy with property x=x0, what s the prob that the other is a boy?" . The probability is prob = ((N/2)-1)/(N-1) where N=2 * 2 * (number of different property values for x). For N>>1, prob tends to 1/2