You visit a town with exactly 1,000 families. Each family has exactly two children. On the street, your tour guide points to a passing man and boy and says, "That's Mr. Smith and his son." With only that information, what is the probability that Mr. Smith's other child is a boy?
user
Sure, I can help with this probability question.

Let's assume that in this town, the gender of a child is determined randomly and independently of other children or families. This is a simplification that makes the math easier.

Since each child is assigned a gender with equal probability (assuming gender is not determined by any other factors in this scenario), the probability that a particular child is a boy is 0.5.

Now, suppose there are 1,000 families in this town. This means there are 2,000 children (each family has two children).

In this town, each child is independently and randomly assigned to be a boy with probability 0.5. Therefore, the probability that exactly one of Mr. Smith's two children is a boy is equal to the probability that the first child is a boy and the second child is not a boy, plus the probability that the first child is not a boy and the second child is a boy.

This can be expressed using a binomial expansion or probability distribution, but a simpler way to answer this question is as follows:

The probability of getting one head in two tosses of a fair coin is 1 x P(heads) + 0 x P(two heads) = 1 x 0.5 + 0 x 0.25 = 0.5. ( coin has two sides heads,tails...)

Similarly, the probability of getting one boy in two children of a family is 1 x 0.5 + 0 x 0.25 = 0.5.

So, the probability that only one of Mr. Smith's children is a boy is 0.5.

Hope that helps! Let me know if you have any other questions.
assistant
You visit a town with exactly 1,000 families. Each family has exactly two children. On the street, your tour guide points to a passing man and boy and says, "That's Mr. Smith and his son." With only that information, what is the probability that Mr. Smith's other child is a boy?
user
The probability that Mr. Smith's other child is a boy is 1/2. This is because the gender of the second child is independent of the gender of the first child. Therefore, the probability remains the same as if you were guessing the gender of any child, which is either a boy or a girl, so 1/2 or 50%.

This question is often mistakenly answered with 13/26 or approximately 0.5, which is derived from the misinterpretation of the boy-girl family combinations in a town with 1,000 families. However, having prior knowledge of one family's boy does not affect the gender probability of subsequent children in that family or the overall population.
assistant