All the families in Dogpatch have exactly two children. For...

All the families in Dogpatch have exactly two children. For these families we can represent the children by bb, bg, gb, gg. In each pair $b$ stands for boy and $g$ for girl; the first letter in each pair represents the older child. We assume boys and girls are equally likely so that probability of each sample point is $1 / 4$. (a) Given that a family has a boy (event B), what is the probability that both children are boys (event A)? (b) Given that the older child is a boy (event C), what is the probability that both children are boys (event A)? (c) Let A be the event that "the family has children of both sexes," and B the event "there is at most one girl." Are A and B independent?

All the families in Dogpatch have exactly two children. For these families we can represent the children by bb, bg, gb, gg. In each pair $b$ stands for boy and $g$ for girl; the first letter in each pair represents the older child. We assume boys and girls are equally likely so that probability of each sample point is $1 / 4$.
(a) Given that a family has a boy (event B), what is the probability that both children are boys (event A)?
(b) Given that the older child is a boy (event C), what is the probability that both children are boys (event A)?
(c) Let A be the event that "the family has children of both sexes," and B the event "there is at most one girl." Are A and B independent?

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Key Concepts

Independence

Two events are independent if the occurrence of one does not influence or alter the probability of the occurrence of the other. This is mathematically expressed as P(A and B) = P(A) * P(B). Establishing the independence or dependence of events is essential in evaluating how combined events interact in probability calculations.

Sample Space and Equally Likely Outcomes

A sample space is the set of all possible outcomes in a probability experiment. When outcomes are equally likely, each outcome is assigned the same probability. This ensures that the probabilities of individual outcomes can be directly used to calculate the likelihood of events by counting favorable outcomes over the total number of outcomes.

Conditional Probability

Conditional probability refers to the likelihood of one event occurring given that another event has already occurred. It is defined by the expression P(A|B) = P(A and B) / P(B), and is crucial for situations where prior information affects the probability of the event in question.

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