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Smith's Children [2] and the Mrs. The initial formulation of the question dates back to at leastwhen Martin Gardner published one of the earliest variants of the paradox in Scientific American.

Titled The Two Children Problemhe phrased the paradox as follows:. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk, [4] and Nickerson. The paradox has frequently stimulated a great deal of controversy. The paradox stems from whether the problem setup is similar for the two questions. The two possible answers share a number of assumptions. First, it is assumed that the space of all possible events can be easily Bou trouble two, providing an extensional definition of Bou trouble two Second, it is assumed that these outcomes are equally probable.

The mathematical outcome would be the same if it were phrased in terms of a coin toss.

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Under the aforementioned assumptions, in this problem, a random family is selected. In this sample space, there are four equally probable events:. Only two of these possible events meet the criteria specified in the question i. This question is identical to question one, except that instead of specifying that the older child is a boy, it is specified that at least one of them is a boy.

In response to reader criticism of the question posed inGardner agreed that a precise formulation of the question is critical to getting different answers for question 1 and 2.

Specifically, Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:. Grinstead and Bou trouble two argue that the question is ambiguous Bou trouble two much the same way Gardner did.

For example, if you see the Bou trouble two in the garden, you may see a boy. The other child may be hidden behind a tree. In this case, the statement is equivalent to the second the child that you can see is a boy. The first statement does not match as one case is one boy, one girl.

Then the girl may be visible. The first statement says that it can be either. While it is certainly true that every possible Mr. Smith has at least one boy i. Smith with at least one boy is intended.

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That is, the problem statement does not say that having a boy is a sufficient condition for Mr. Smith to be identified as having a boy this way. Smith, unlike the reader, is presumably aware of the sex of both of his children when making this statement", i.

If it is assumed that this information was obtained by looking at both children to see if there is at Bou trouble two one boy, the condition is both necessary and sufficient. Three of the four equally probable events for a two-child family Bou trouble two the sample space above meet the condition, as in this table:.

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However, Bou trouble two the family was first selected and then a random, true statement was made about the sex of one child in that family, whether or not both were considered, the correct way to calculate the conditional probability is not to count all of the cases that include a child with that sex. Instead, one must consider only the probabilities where the statement will be made in each case. So, if you are told that at least one is a boy when the fact is chosen randomly, the probability that Bou trouble two are boys is.

The paradox occurs when it is not known how the statement "at least one is a boy" was generated. Either answer could be correct, based on what is assumed.

As Marks and Smith say, "This extreme assumption is never included in the presentation of the two-child problem, however, and is surely not what people have in mind Bou trouble two they present it. Another way to analyse the ambiguity for question 2 is by making explicit the generative process all draws are independent.

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Following classical probability arguments, we consider a large urn containing two children. We assume equal probability that either is a boy or a girl. The three discernible cases are thus: These are the prior probabilities. Bou trouble two Bayes' Theoremwe find. The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky.

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Just how do we know that "at least" one is a boy? One description of the problem states that we look into a window, see only one child and it is a boy. This sounds like the same assumption. However, this one is equivalent to "sampling" the distribution i.

Let's call the statement "the sample is a boy" proposition "b". The difference here is the P bwhich is just the probability of drawing a boy from all possible cases i.

The Bayesian analysis generalizes easily to the case in Bou trouble two we relax the If we have no information about the populations then we assume a "flat prior", i. Suppose you had wagered that Mr Smith had two boys, and received fair odds. We think of your wager as investment that will increase in value as Bou trouble two news arrives.

What evidence would make you happier about your investment?

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Learning that at least one child out of two is a boy, or learning that at Bou trouble two one child out of one is a boy? The latter is a priori less likely, and therefore better news.

That is why the two answers cannot be the same. Now for the numbers. If we bet on one child and win, the value of your investment has doubled. On the other hand if we learn that at least one of two children is a boy, our investment increases as if we had wagered on this question.

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So the answer is 1 in 3. Following the popularization of the paradox by Gardner it has been presented and discussed in various forms.

However, someone may argue that "…before Mr. Smith identifies the boy as his son, we know only that he is either the father of two boys, BB, or of two girls, GG, or of one of each in either birth order, i. Discovering that he has at least one boy rules out the event GG. The natural assumption is that Mr. Smith selected the child companion at random. They imagine a culture in which boys are invariably chosen over girls as walking companions. Bou trouble two

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InMarilyn vos Savant responded to a reader who asked her to answer a variant of the Boy or Girl paradox that included beagles.

The and questions, respectively were phrased:. In response to reader response that questioned her analysis vos Savant conducted a survey Bou trouble two readers with exactly two children, at least one of which is a boy.

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Of 17, responses, The authors do not discuss the possible ambiguity in the question and conclude that her answer is correct from a mathematical perspective, given the assumptions that the likelihood of a child being a boy or girl is equal, and that the sex of the second child is independent of the first.

Carlton and Stansfield go on to discuss the common assumptions in the Boy or Girl paradox. They demonstrate that in reality male children are actually more likely than female children, and that the sex of the second child is not independent of the sex of the first. The authors conclude that, although the assumptions of the question run counter to observations, Bou trouble two paradox still has pedagogical value, since it "illustrates one of the more intriguing applications of conditional probability.

Suppose we were told not only that Mr. Smith has two children, and one of them is a Bou trouble two, but also that the boy was Bou trouble two on a Tuesday: Again, the answer depends on how this information was presented - what kind of selection process produced this knowledge. Following the tradition of the problem, suppose that in the population of two-child families, the sex of the two children is independent of one another, equally likely boy or girl, and that the birth date of each child is independent of the other child.

From Bayes' Theorem that the probability of two boys, given that one boy was born on a Tuesday is given by:. For the denominator, let us decompose: The first term is already known by the previous remark, the last term is 0 there are no boys.

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Therefore, the full equation is:. In other words, as more and more details about the boy child are given for instance: To understand why this is, imagine Marilyn vos Savant's poll of readers had asked which Bou trouble two of the week boys in the family were born.

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If Marilyn then divided the whole data set into seven groups - one for each day of the week a son was born - six out of seven families with two boys would be counted in two groups the group for the day of the week of Bou trouble two boy 1, and the group of Bou trouble two day of the week of birth for boy 2doubling, in every group, the probability of a boy-boy combination. However, is it really plausible that the family with at least one boy born on a Tuesday was produced by choosing just one of such families at random?

It is much more easy to imagine the following scenario. Assume that which of the two children Bou trouble two the door is determined by chance. Then the procedure was 1 pick a two-child family at random from all two-child families 2 pick one of the two children at random, 3 see if it is a Bou trouble two and ask on what day he was born. This is a very different procedure from 1 picking a two-child family at random from all families with two children, at least one a boy, born on a Tuesday.

This variant of the boy and girl problem is discussed on many internet blogs and is the subject of a paper by Ruma Falk. However, this does not exhaust the boy or girl paradox for it is not necessarily the ambiguity that explains how the intuitive probability is derived.

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Ambiguity notwithstanding, this makes the problem of interest to psychological researchers who seek to understand how humans estimate probability. Smith problemcredited to Gardner, but not worded exactly the same as Gardner's version to test theories of how people estimate conditional probabilities.

The authors argued that the reason people respond differently to each question along with other similar problems, such Bou trouble two the Monty Hall Problem and the Bertrand's box paradox is because of the use of naive heuristics that fail to properly define the number of possible outcomes.

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