Consider a family with 3 children. Assume the probability...

Consider a family with 3 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also $0.5,$ and that the events "boy" and "girl" are independent. (a) List the equally likely events for the gender of the 3 children, from oldest to youngest. (b) What is the probability that all 3 children are male? Notice that the complement of the event "all three children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female.

Consider a family with 3 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also
$0.5,$ and that the events "boy" and "girl" are independent.
(a) List the equally likely events for the gender of the 3 children, from oldest
to youngest.
(b) What is the probability that all 3 children are male? Notice that the complement of the event "all three children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female.

Key Concepts

Complement Rule

The complement rule is a strategy used in probability to find the likelihood of an event by subtracting the probability of its complement from one. This is particularly useful when calculating the probability of 'at least one' occurrence by first finding the probability of the event not occurring at all.

Multiplication Rule for Independent Events

The multiplication rule is applied when we need to find the probability of multiple independent events occurring together. It states that the overall probability is the product of the probabilities of each individual event, which is crucial for calculating the probability of all children being of a specific gender.

Equally Likely Outcomes

When outcomes in the sample space have the same probability of occurring, they are considered equally likely. This assumption simplifies probability calculations since each event in the sample space contributes equally to the overall probability.

Sample Space

The sample space is the collection of all possible outcomes for a probabilistic experiment. In the context of multi-event scenarios like determining the genders of children, the sample space helps enumerate all potential outcomes, ensuring that every possibility is considered when calculating probabilities.

Independent Events

Independent events are those whose outcomes do not influence each other. In probability theory, the gender of one child has no effect on the gender of another, which allows us to treat each event separately and use independent probability rules.

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