Nobody Understands Probability • Hacker News Board

Re: Nobody Understands Probability

#61

Post by thwarted » Wed, Sep 15, 2010, 3:03 AM UTC

jodrellblank wrote: post #59

thwarted wrote: post #48

Earlier quoted context omitted.

I didn't "forget" about it, I contend that case is already handled by GB; "one child is a boy and one child is a girl" is the same as "one child is a girl and one child is a boy" and can be expressed as either GB or BG, making GB and BG equivalent.

GB and BG take up more area in probability space. If you have a thousand paths in front of you, one leading to a fortune, one to a potion and the rest to a pit of death, you can't say the 998 paths are together equal to the other two just because they all end at the same place. It's much more likely you'll hit the pit of death. It's much more likely to have boy/girl children even though BG and GB look the same at the…

If there are four buildings, red, green, gray and gray, and you pick one at random you are more likely to end up in a gray building.

I guess that's where my confusion lies. I see that the chances of picking a grey building the above is 1/2, but I assert that the set of possibilities for the children is "two girls", "different genders" and "two boys". But yours is a good explanation, I see that that "different genders" state has a double chance of occurring compared to the other two states.

The Wikipedia article I cited above is interesting in that in one explanation, it bases the different results on interpretation of the givens. I see the problem stated as about a specific man's state, not the chances that a randomly chosen man/family with two children has two boys. In my case, this was a misdirection in trying to understand this.

Re: Nobody Understands Probability

#62

Post by Tichy » Wed, Sep 15, 2010, 6:56 AM UTC

Dn_Ab wrote: post #60

Tichy wrote: post #56

Earlier quoted context omitted.

I see - I thought the first child you meet is always a boy in your example. So I am still not sure what you are calculating, but at least I understand the tuple notation :-)

I was trying to make a point that two problems that look the same actually lead to different results based on what information you are given. For example, being told someone has 2 kids and at least one is a girl then the space for this is S = {(M,F), (F,M), (F,F)}. However if I were to run into someone with two kids and saw a daughter then the information I have doesn't allow me to construct the space S above if I wa…

I must admit, I don't understand the distinction. What does age have to do with it? The information seems to be the same in both cases: a man with two kids, at least one of them a girl.

Of course thinking about age is a legit way to calculate the probability of the second kid being a girl. It is just unnecessarily complicated.

Re: Nobody Understands Probability

#63

Post by tel » Wed, Sep 15, 2010, 5:06 PM UTC

iliketosleep wrote: post #36

tel wrote: post #35

Earlier quoted context omitted.

In a sampling of 1000 families, the expected values of each kind of family is as follows: 2xB : 250 1xB, 1xG: 500 2xG : 250 Sampling this population ignoring any family that has no boys leads to the probabilities 2xB : 1/3 1xB, 1xG: 2/3rds You're still looking at the same probabilities; the models agree. I don't fully understand your two random variables formulation. I think the confusion you're getting at is that th…

ok, you stated this very well. It's now clear where the confusions arises: "Sampling this population ignoring any family that has no boys." Yes, with this interpretation the answer is 1/3, but it's contrary to my interpretation. Actually, for anyone who is interested, see "Boy or Girl paradox". There is literature on this which discusses the different interpretations.

That sampling arises because being part of the population which has no boys is * necessary and sufficient* to (truthfully) make the statement that forms the paradox.

The alternative interpretation of the paradox arises when the wording of the paradox is construed to identify one of the children as male or female. In this case (stating something like "my first child is male"), being part of the population (x \in {BB, BG}) is necessary and sufficient and leads to the 1/2 probability of having two boys.

In short, the question becomes whether you believe the child is identified in the wording of the question. Honestly, the author of the paradox goes pretty far out of their way to say "at least one of the children is male" avoiding that identification.

Re: Nobody Understands Probability

#64

Post by Dn_Ab » Thu, Sep 16, 2010, 12:50 AM UTC

Tichy wrote: post #62

Dn_Ab wrote: post #60

Earlier quoted context omitted.

I was trying to make a point that two problems that look the same actually lead to different results based on what information you are given. For example, being told someone has 2 kids and at least one is a girl then the space for this is S = {(M,F), (F,M), (F,F)}. However if I were to run into someone with two kids and saw a daughter then the information I have doesn't allow me to construct the space S above if I wa…

I must admit, I don't understand the distinction. What does age have to do with it? The information seems to be the same in both cases: a man with two kids, at least one of them a girl. Of course thinking about age is a legit way to calculate the probability of the second kid being a girl. It is just unnecessarily complicated.

The distinction is in how the information was presented and there is a difference (let us ignore the ages and assume i met the younger one then seeing as I met one girl my space is {(F,F), (F,M)} vs {(F,F), (F,M), (M,F)} for at least one girl while accounting for age). For the second age does not matter so much as who is older and accounting for all combinations is why I present it that way. The distinction is tricky but problems like these are often covered in the conditional probabilities section of most probability theory texts.