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It was just a little — and then everything was normal. Darya insists she has had no other partner. The girl plans to keep her baby. The girl and her mother want to keep the baby, but it is too early in the pregnancy for a DNA check on the child, which will be completed later.

This article originally appeared on The Sun and was reproduced with permission. Log in Sign up. Log out. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk, [4] and Nickerson.

The paradox has 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. 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.

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 in , Gardner 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 Snell argue that the question is ambiguous in much the same way Gardner did. For example, if an observer sees the children in the garden, they 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 the observer 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. 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 least one boy, the condition is both necessary and sufficient.

Three of the four equally probable events for a two-child family in the sample space above meet the condition, as in this table:.

However, if 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 at least one is a boy when the fact is chosen randomly, the probability that both 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 when they present it.

Another way to analyse the ambiguity for question 2 is by making explicit the generative process all draws are independent.

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: 1. These are the prior probabilities. Using Bayes' Theorem , we 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.

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.

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Two young couples met in a youth hostel. Women use smartphones to take selfies and view messages; men are bored. Despite widespread reports that Ivan made Darya pregnant, the popular On Air Live show with presenter Andrey Malakhov brought on Dr Evgeny Grekov, a urology and andrology expert, who denied Ivan could be the father.

And just to add, he still has childish sex organs. Puberty has not started. So we have a lot of questions.

The Rossiya 1 channel show — called Father at 10!? They just want to hang it on the boy now. A doctor has ruled Ivan is still too young to produce sperm.

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".

Now we have:. The difference here is the P b , which is just the probability of drawing a boy from all possible cases i.

The Bayesian analysis generalizes easily to the case in which we relax the population assumption. If we have no information about the populations then we assume a "flat prior", i.

Suppose one had wagered that Mr. Smith had two boys, and received fair odds. Their wager will increase in value as good news arrives.

What evidence would make them happier about their investment? Learning that at least one child out of two is a boy, or learning that at least 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 their investment has doubled. On the other hand if one were learn that at least one of two children is a boy, the investment increases as if they had wagered on this question.

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.

In , Marilyn 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 of readers with exactly two children, at least one of which is a boy.

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, the 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 boy, but also that the boy was born on a Tuesday: does this change the previous analyses?

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:. The first term is already known by the previous remark, the last term is 0 there are no boys.

Therefore, the full equation is:.

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