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September 4

12 people in a family, all with birthdays in different months

Say there's a family of 12 people: 2 parents and 10 children. I'm told all the kids came from single births. Someone claims that none of the 12 have a birthday in the same month as any of the others, i.e. there's exactly one birthday for each of the 12 months. How likely is this claim to be true? -- JackofOz (talk) 21:26, 4 September 2009 (UTC)[reply]

As a rough estimate if you took 12 random people and assumed randomly distributed birthdays among the 12 months (which they aren't), the probability of them having one birthday per month is 12!/12¹², which is about 1 in 19000. That's not to say it's a lie. Unusual things do happen some of the time, and when they do there's typically disproportionate attention brought to them, which can make them seem like they happen more often then they rightfully should. I think there's probably a name for that effect but I don't know what it is. Rckrone (talk) 21:52, 4 September 2009 (UTC)[reply]

That is "how likely it is to be true" if the family were chosen uniformly from among all families with 12 children.

Rckrone wrote:

assumed randomly distributed birthdays among the 12 months (which they aren't)

Here's another instance of the use of "randomly" to mean "uniformly". This is an oddly persistent meme. I don't know why. Michael Hardy (talk) 22:09, 4 September 2009 (UTC)[reply]

What I meant to say was that 1/19000 is a rough estimate of the probability of this happening among 12 random people. I agree that the probability that the claim is true is much much higher. In fact I don't see any reason to think it isn't. As for misusing "randomly" I have no explanation. Rckrone (talk) 22:17, 4 September 2009 (UTC)[reply]

Suppose there are n families with 2 parents and exactly 10 children, and the 12 birthdays are uniformly distributed in the 12 months. The probability of at least one occurrence of one birthday for each of the 12 months is 1 - (1-12!/12¹²)ⁿ = 1 - 0.999946276783ⁿ. This passes 50% at n = 12902, and it is 99.999999999999999999999% at n = 1000000. I don't know how many such families there are and how far from uniform distribution the birthdays really are but it seems highly likely that there exists such an occurrence. PrimeHunter (talk) 23:33, 4 September 2009 (UTC)[reply]

Oh, I have no doubt it has happened many times, because it's obviously well within the realms of possibility. But if I were a betting man, and someone asserted this particular family had this characteristic, I was wanting to know whether it would be a better bet to lay money they were telling the truth, or they were telling a lie (or, to be charitable, they were mistaken). That may not be exactly what I originally asked, but it's what I meant. -- JackofOz (talk) 23:43, 4 September 2009 (UTC)[reply]

It would depend on the reliability of the person making the claim. If they knew the family and I didn't then I would be reluctant to offer a bet to them. If they accept the bet then it may be because they feel fairly confident they are right. Almost everybody knows some things which are rare, and rare things often draw attention and may be told many times. PrimeHunter (talk) 00:13, 5 September 2009 (UTC)[reply]

"the use of "randomly" to mean "uniformly". ... I don't know why". The article randomness explains that the word "random" refers to a discrete uniform distribution: "Governed by or involving equal chances for each of the actual or hypothetical members of a population". Mathematicians change the meaning of words to include special cases, such that a coin with two tails still produces a random variable even if the outcome is neither random nor variable, but predictable and constant. Bo Jacoby (talk) 06:43, 5 September 2009 (UTC).[reply]

The 12!/12¹² calculation assumes that the birthdays are independent of one another. I'd say that it would be hard work to establish/defend any statement of independence here. Robinh (talk) 09:20, 5 September 2009 (UTC)[reply]

That's certainly true (and even the assumption that the distribution is uniform measured by months is weird and not supported by the principle of indifference -- we know, after all, that some months are longer than others). But as a practical matter, to get a first approximation without doing too much work, it's a reasonable guess. --Trovatore (talk) 20:59, 5 September 2009 (UTC)[reply]

For a busy family like that, independence is the last thing I'd assume. Mom gets pregnant in month X, kid #1 is born 9 months later, Mom spends 2 months nursing and getting her parts back to normal, and at month X+11 gets pregnant again. Kid #2 is born at X+20, rinse, repeat. A new pregnancy every 11 months, and since 11 (months between pregnancy starts) and 12 (months in a year) are relatively prime, you get all the births in distinct months. See also Cheaper by the Dozen. 67.122.211.205 (talk) 07:49, 6 September 2009 (UTC)[reply]

"Uniformly distributed" does not mean the same as discrete uniform distribution to most readers. Surely "randomly distributed" is less likely to be misinterpreted? Both are unlikely in these circumstances. Dbfirs 09:56, 6 September 2009 (UTC)[reply]

Here's a point that nobody mentioned. What is the likelihood of conscious manipulation of the birth months by the parents? Some fertile couples might make it a point to create such a family, and then all purely mathematical considerations are totally out the window (which they probably should have been from the beginning on other grounds). It makes a nice little exercise if you stipulate a uniform distribution, but another question should be put in its place unless you want a very complex mostly non-mathematical discussion.Julzes (talk) 01:06, 8 September 2009 (UTC)[reply]

Canonical Forms for PDEs

Okay so working with 2nd order linear PDEs, I cam across the fact that there are canonical forms for different kinds of PDEs. For hyperbolic PDEs, the canonical forms are $u_{xx}-u_{tt}=f(x,t,u,u_{x},u_{t})$ or $u_{xt}=g(x,t,u,u_{x},u_{t})$ . My question is why are there two canonical forms for hyperbolic PDEs? Is there a significance to this distinction? Thinking with analogies to conics sections, I think that we just "rotate" one form by 45 degrees to get another form. Why do we do that? Is one easier to solve in some cases than the other?--97.118.56.41 (talk) 22:37, 4 September 2009 (UTC)[reply]

This is a very nice question. There are some papers by Farid Tari, and some other co-authors, that address the classifications of implicit differential equations.

"Bifurcations of binary differential equations." Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 485–506. (With J. W. Bruce and G. J. Fletcher)
"Duality and implicit differential equations", Nonlinearity 13 (2000), 791–812. (With J. W. Bruce)
"Two-parameter families of implicit differential equations", Discrete Contin. Dynam. Systems, 13 (2005), 139–262.
"Two-parameter families of binary differential equations", Discrete Contin. Dynam. Systems, 22 (2008), 759–789.

Maybe these will help ~~ Dr Dec (Talk) ~~ 09:57, 5 September 2009 (UTC)[reply]

It's nothing special. As you know, both form are very common in applications. E.g. as you know, waves and oscillations are described by the first one. As you know, $u_{tx}=0$ is easier to solve, in fact immediate. And, as you know, one can easily pass from one form to the other. So, if you already know the answer, why are you asking? ;) (@Dec: the OP's eq. are semilinear hyperbolic equations, there's nothing implicit!)--pma (talk) 18:18, 5 September 2009 (UTC)[reply]

No, but it's a question of classification. Such classifications always follow similar lines. I thought it might have been of some interest to him/her. ~~ Dr Dec (Talk) ~~ 20:24, 5 September 2009 (UTC)[reply]