Probability of a Girl Assuming that boys and girls are equally li...

4. Probability

Multiplication Rule: Independent Events

Statistics

4. Probability

Multiplication Rule: Independent Events

2:05 minutes

Problem 4.3.6

Textbook Question

Probability of a Girl Assuming that boys and girls are equally likely, find the probability of a couple having a boy when their third child is born, given that the first two children were both girls.

Verified step by step guidance

Step 1: Understand the problem. The question asks for the probability of the third child being a boy, given that the first two children are girls. Note that the genders of the children are independent events, meaning the outcome of one child does not affect the outcome of another.

Step 2: Recall the basic probability rule. When boys and girls are equally likely, the probability of having a boy or a girl for any single child is 1/2 (or 0.5).

Step 3: Recognize that the condition provided (the first two children being girls) does not influence the probability of the third child’s gender. This is because the events are independent.

Step 4: Write the probability of the third child being a boy as P(Boy) = 1/2. This is based solely on the fact that boys and girls are equally likely, and the previous outcomes do not affect this probability.

Step 5: Conclude that the probability of the third child being a boy remains 1/2, regardless of the genders of the first two children. This is a key concept in understanding independent events in probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In this scenario, we are interested in the probability of having a boy as the third child, given that the first two children were girls. This concept is crucial for understanding how prior outcomes can influence the probability of future events.

Independence of Events

In probability, two events are considered independent if the occurrence of one does not affect the occurrence of the other. In this case, the gender of the third child is independent of the genders of the first two children. This means that regardless of the first two being girls, the probability of the third child being a boy remains unchanged.

Sample Space

The sample space is the set of all possible outcomes of a probabilistic experiment. For the birth of children, the sample space consists of combinations of boys and girls. In this problem, the relevant outcomes for the third child are 'boy' or 'girl', and understanding the sample space helps clarify the probabilities associated with each outcome.

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