The Tuesday boy problem.
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From: Ostap Bender on 2 Jun 2010 22:19 On Jun 2, 7:08 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-02, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > > > Here is more information for you: I have two children, and one of them > > is a boy born on a Tuesday who has brown hair, reads comic books, and > > is left-handed. > > > Is that going to change your probabilities? > > For the three examples of fairly reasonable interpretations I posted > previously in this subthread: no, no, and yes respectively. > > What do you expect with an ambiguous question? What do **I** expect? I expect most people here to be familiar with the classic Martin Gardner paradox which I first read in 7th grade and solved in 8th grade: http://en.wikipedia.org/wiki/Boy_or_Girl_paradox Boy or Girl paradox The Boy or Girl paradox surrounds a well-known set of questions in probability theory which are also known as The Two Child Problem[1], Mr. Smith's Children[2] and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner published one of the earliest variants of the paradox in Scientific American. Titled The Two Children Problem, he phrased the paradox as follows: * Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls? * Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys? Gardner initially gave the answers 1/2 and 1/3, respectively; but later acknowledged[1] that the second question was ambiguous. Its answer could be 1/2, depending on how you found out that one child was a boy. --------- Thus, the answer can be either 1/2 or 1/3, depending on how the answer was selected. But under all reasonable interpretations, it cannot be anything else, and it doesn't depend on the birth day, nor on left- handedness, nor on any other properties of the reported boy.
From: Bastian Erdnuess on 2 Jun 2010 22:22 John L. Barber wrote: >> John L. Barber wrote: >> >> > I'd just like to mention that I've just done a >> > small simulation of this process. [...] >> >> > I keep track of Ntb, the total number of pairs >> > so far which have included at least one Tuesday >> > Boy, as well as Nbb_tb, the number of pairs >> > containing at least one Tuesday Boy which also >> > contain two boys. >> >> How do you see from the OP that you really get >> informed about the Tuesday Boy in anycase where >> there is a Tuesday Boy in the family? > > It's not explicitly stated. > > However, this type of problem is of a certain class > of problems (the archetype being the Monty Hall > problem) in which the same types of assumptions are > always made. In the Monty Hall problem it is explicitly stated how the quiz master chooses the first door to open. And that's important! > The opening poster knew that the people on a forum > like this one, being math-game-oriented folks, > would be familiar with this kind of problem, in > which the "you always get informed" assumption is > always implicit. I think you missed the point about the "always get informed" thing. I think it is easier to work out in the other question I posted. You could also have a look at hangman's posting and ask yourself what makes you beleve that you are in case (a). > In other words, it is assumed that the readers know > the basic ground rules that invariably accompany > such puzzles. And that's why he writes: "Don't be too quick to react. Just as with Monty Hall some well-known and well-respected mathematicians are quite definite, it's just that they don't all get the same answer." ? Btw. even Martin Gardner (author of the Mr. Smith problem, rip) agreed later that his question in the original form was not unambiguous[1]. For the very same reason Gary Foshee's question is not either. Cheers, Bastian [1] see e.g. <http://en.wikipedia.org/wiki/Boy_or_Girl_paradox#Ambiguous_problem_statements>
From: Ostap Bender on 2 Jun 2010 22:37 On Jun 2, 7:19 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > On Jun 2, 7:08 pm, Tim Little <t...(a)little-possums.net> wrote: > > > On 2010-06-02, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > > > > Here is more information for you: I have two children, and one of them > > > is a boy born on a Tuesday who has brown hair, reads comic books, and > > > is left-handed. > > > > Is that going to change your probabilities? > > > For the three examples of fairly reasonable interpretations I posted > > previously in this subthread: no, no, and yes respectively. > > > What do you expect with an ambiguous question? > > What do **I** expect? I expect most people here to be familiar with > the classic Martin Gardner paradox which I first read in 7th grade and > solved in 8th grade: > > http://en.wikipedia.org/wiki/Boy_or_Girl_paradox > > Boy or Girl paradox > > The Boy or Girl paradox surrounds a well-known set of questions in > probability theory which are also known as The Two Child Problem[1], > Mr. Smith's Children[2] and the Mrs. Smith Problem. The initial > formulation of the question dates back to at least 1959, when Martin > Gardner published one of the earliest variants of the paradox in > Scientific American. Titled The Two Children Problem, he phrased the > paradox as follows: > > * Mr. Jones has two children. The older child is a girl. What is > the probability that both children are girls? > * Mr. Smith has two children. At least one of them is a boy. What > is the probability that both children are boys? > > Gardner initially gave the answers 1/2 and 1/3, respectively; but > later acknowledged[1] that the second question was ambiguous. Its > answer could be 1/2, depending on how you found out that one child was > a boy. > --------- > > Thus, the answer can be either 1/2 or 1/3, depending on how the answer > was selected. But under all reasonable interpretations, it cannot be > anything else, and it doesn't depend on the birth day, nor on left- > handedness, nor on any other properties of the reported boy. Wait! I didn't give it much thought, did I? So, your point is that there can be more than two ways to interpret this? You are probably right. I was wrong. Sorry. But why would you say that it can depend on whether he is left- handed, but doesn't depend on whether his hair is brown, or that he reads comic books? What if I told you that Pr{hair=Brown} = 30%, Pr{hair=notBrown} = 70% Pr{readsComix=True}=50%, Pr{readsComix=False}=50% Pr{handedness=Left} = 10%, Pr{handedness=Right} = 90% and all these r.v's are independent of each other and of the gender? What makes the effect of handedness different from hair color?
From: Ostap Bender on 2 Jun 2010 22:54 On May 26, 12:28 pm, riderofgiraffes <mathforum.org...(a)solipsys.co.uk> wrote: > A relative of the Monty Hall problem, and currently > causing some division and argument. Originally posed > by Gary Foshee. > > Suppose I have two children, and one of them is a boy > born on a Tuesday. What is the probability that both > my children are boys? The answer depends on your priors. For example when you say "one of them is a boy born on a Tuesday", you could mean different things: 1. At least one of them is a boy born on a Tuesday 2. At least one of them is a boy, and he was born on a Tuesday 3. The first child that I lay my eyes on, is a a boy born on a Tuesday Etc. I think the answers are 13/27, 1/3, and 1/2 respectively. > Don't be too quick to react. Just as with Monty Hall > some well-known and well-respected mathematicians are > quite definite, it's just that they don't all get the > same answer.
From: Bastian Erdnuess on 2 Jun 2010 23:10
Ostap Bender wrote: >> > > Here is more information for you: I have two children, and one of them >> > > is a boy born on a Tuesday who has brown hair, reads comic books, and >> > > is left-handed. >> >> > > Is that going to change your probabilities? >> >> > For the three examples of fairly reasonable interpretations I posted ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ >> > previously in this subthread: no, no, and yes respectively. > But why would you say that it can depend on whether he is left- > handed, but doesn't depend on whether his hair is brown, or that he > reads comic books? It's not what he is saying. The "no, no, yes" does not refer to "brown, comic, left" but to the "three examples of fairly reasonable interpretations" he postend in his OP. Cheers, Bastian |