Wikipedia:Reference desk/Archives/Mathematics/2008 April 18

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[edit] April 18

[edit] Really confusing probability question

I read this question in some blog that talked about the Monty Hall 3-door problem: "1. Mr. Smith has two children, at least one of whom is a boy. What is the probability that the other is a boy?" Apparently the answer is 1/3, because "Three equally likely combinations are BG, GB and BB, and only one of them has the (rather ill-defined) “other” child being a boy." But it makes no sense to me. Consider that the second child is as of yet not conceived when the question is asked. Its chance of coming into the world as a boy should be entirely independent of its sibling's sex. And yet this question suggests that somehow one is dependent on the other. Is this really right? Can somebody explain this to me in a convincing way that does not use those permutation possibilities (cause I just can't buy them.)? Thanks, 140.247.237.63 (talk) 04:47, 18 April 2008 (UTC)

We have an article on this actually, Boy_or_Girl. What you've pointed out is in fact a subtle distinction which actually affects the problem. Counter-intuitiveness aside, the explanation on that page is quite solid. Someletters<Talk> 05:01, 18 April 2008 (UTC)
The issue is the conditional probability involved. In particular, it's about how much you know about the situation. If you only know that at least one child is a boy, but don't know whether it's the eldest or youngest, then that's when you get the 1/3 case. If you know that the eldest is a boy, then of course the sex of the youngest is independent of that. Imagine 100 random families, each with two children. Now of those 100, roughly 25 will have two girls, 25 will have two boys, and 50 will have one of each. Hence, 75 families have at least one boy, and of those, 25 have two, giving you 25/75 = 1/3 probability that a family has two boys, given that it is known it has at least one. Confusing Manifestation(Say hi!) 05:08, 18 April 2008 (UTC)

Thanks! 140.247.237.63 (talk) 06:52, 18 April 2008 (UTC)

[edit] Reversing Normal Distribution Function

I have the normal distribution function as follows:

\phi \left( Z \right)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{Z}{e^{-\frac{1}{2}t^{2}}dt}

I'm trying to rearrange it so that Z is the subject. However, I cannot work out how to integrate the above (which I assume I need to) in order to reverse the equation. The nearest I've seen is the integral of e^{x^{2}}, which has a long result, though it's still not the one I want. So I was wondering if anyone knew how to integrate the above? Robert (talk) 13:07, 18 April 2008 (UTC)

If by "integrate" you mean "find an antiderivative for", you're out of luck. e^{-\frac{1}{2}t^{2}} has no elementary antiderivative. –King Bee (τ • γ) 13:09, 18 April 2008 (UTC)
Well, you can always write it in terms of \mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt (though that might not be of great help, because it's not an elementary function as said King Bee)
Letting u2 = ½ x2, \int e^{-\frac{1}{2}x^2} dx = \int e^{-u^2} \cdot \sqrt{2} du = \frac{\sqrt{\pi}}{2} \cdot \sqrt{2} \cdot \mbox{erf}(u) = \frac{\sqrt{\pi}}{2} \cdot \sqrt{2} \cdot \mbox{erf} \left(\frac{\sqrt{2}x}{2} \right) -- Xedi (talk) 14:30, 18 April 2008 (UTC)
The rephrase what the others have said - you can't integrate it algebraically, you have to do it numerically. That's why you have look up tables for the normal distribution. --Tango (talk) 15:45, 18 April 2008 (UTC)

[edit] In algebra real numbers the only completely ordered field?

The article Number says:

In abstract algebra, the real numbers are uniquely characterized by being the only completely ordered field. They are not, however, an algebraically closed field.

The article Total order says:

if X is totally ordered under ≤, then the following statements hold for all a, b and c in X:
If a ≤ b and b ≤ a then a = b (antisymmetry);
If a ≤ b and b ≤ c then a ≤ c (transitivity);
a ≤ b or b ≤ a (totality or completeness).

Would this not also apply to Integers and rational numbers? -- Q Chris (talk) 14:49, 18 April 2008 (UTC)

Well, the integers aren't a field (there are no multiplicative inverses except for 1 and -1). I'm not sure why the rational numbers don't count - by my understanding, they are a totally ordered field. They're not complete, in the analytical sense, but I don't think that's what it's talking about. --Tango (talk) 15:41, 18 April 2008 (UTC)
There is indeed an error in the article Number. The Reals are the only (up to isomorphy) complete ordered field (not the missing -ly). This changes the meaning of this phrase. "Ordered field" means "totally ordered field", i.e. a field being totally ordered as explained at Total order. The "complete" does note modify "ordered" but adds the property of Completeness (order theory) to the phrase. This is where the rationals fail. The article Real numbers has it correct. —Preceding unsigned comment added by 80.130.141.245 (talk) 15:57, 18 April 2008 (UTC)
An ordered field is a more narrow concept than that of a field with a total order on it. The order has to respect the field operations. The complex numbers with the lexicographical ordering is Cauchy complete and has a total order, but it is not an ordered field. 134.173.93.127 (talk) 05:33, 19 April 2008 (UTC)
Now fixed in Number 80.130.141.245 (talk) 15:59, 18 April 2008 (UTC)
Actually, the reals aren't complete in the sense of order theory. They're what our article calls bounded complete, and is sometimes called Dedekind complete. This is distinct from the property of being Cauchy complete (which the reals also have); there are lots of nonisomorphic Cauchy complete ordered fields. Algebraist 00:13, 19 April 2008 (UTC)

[edit] number line

Which of the following statements about the number line is true?

A. Number lines end with the number 10.
B. Numbers get smaller as you move to the left on the number line.
C. Number lines can't help you compare numerals.
D. Numbers get larger as you move to the left on the number line
—Preceding unsigned comment added by 67.55.21.71 (talk • contribs) 18:51, 18 April 2008

E. People who post homework questions should be ignored. --LarryMac | Talk 18:57, 18 April 2008 (UTC)
We are glad to help with homework, but only if you do your part. Tell us what answer you think is correct, or at least which answers you can eliminate, and we will tell you if we agree. One hint is that B and D appear to say the opposite of each other, so, if one is true then the other is false. StuRat (talk)
Please see our Number line article. --hydnjo talk 22:50, 18 April 2008 (UTC)