Question

In a family consisting of a mother, father, and two children of different ages, what is the probability that the family has two girls, given that one of the children is a girl? What is the probability that the children are both boys, given that the older child is a boy?

Short answer

Probability of two girls given one is a girl: \(\frac{1}{3}\). Probability of both boys given the older is a boy: \(\frac{1}{2}\).

Step-by-step solution

Define the Sample Space

By considering two children, the possible gender combinations are: Boy-Boy (BB), Boy-Girl (BG), Girl-Boy (GB), and Girl-Girl (GG). Each combination is equally likely.

Identify Favorable Outcomes for First Scenario

For the scenario where one child is a girl, the possible outcomes are BG, GB, and GG. We need to find the probability of GG given that one child is a girl.

Calculate Probability for First Scenario

The possible outcomes given that one child is a girl are BG, GB, GG (3 outcomes). The favorable outcome (two girls) is GG (1 outcome). Probability = \(\frac{1}{3}\).

Identify Favorable Outcomes for Second Scenario

For the older child being a boy, the relevant outcomes are BB and BG. We need to find the probability of BB given that the older child is a boy.

Calculate Probability for Second Scenario

The possible outcomes given that the older child is a boy are BB and BG (2 outcomes). The favorable outcome (both boys) is BB (1 outcome). Probability = \(\frac{1}{2}\).

Key concepts

Sample Space

Understanding the concept of a sample space is essential in probability theory. A sample space represents all the possible outcomes of a particular experiment or situation. In the context of our exercise, the family setup with two children presents a few combinations of genders. Each child can either be a boy or a girl, leading to these possibilities:

• Boy-Boy (BB)
• Boy-Girl (BG)
• Girl-Boy (GB)
• Girl-Girl (GG)

This list shows the complete sample space for our exercise. Each combination of children is an outcome that could occur, considering the condition that the gender assignments are independent and equally likely. Grasping the sample space concept helps us further explore how to determine probabilities for specific conditions.

Probability Theory

Probability theory is the mathematical framework for gauging how likely events are to occur. In practical terms, it helps us assess situations and predict outcomes. Given our family example, we apply this theory to find the probability of certain family compositions, depending on specific known factors.When finding the probability that the family has two girls given one of the children is a girl, we identify all scenarios fitting that condition. In this case, the possible outcomes are:

• Boy-Girl (BG)
• Girl-Boy (GB)
• Girl-Girl (GG)

Here, probability is calculated by looking at favorable and possible outcomes. Probability = (Favorable Outcomes)/(Total Possible Outcomes). For the two-girls scenario with one known girl, the probability is \(rac{1}{3}\).This instance highlights using probability theory by refining our sample space based on conditions shared.

Favorable Outcomes

Favorable outcomes are those specific occurrences that align with the event we're interested in. Identifying these is crucial for calculating probability accurately. For each scenario, we filter out the outcomes from the sample space that fit the condition specified.Take the first scenario, where we need to find the probability of having two girls (GG), given at least one child is a girl. Among possible outcomes (BG, GB, GG), the favorable one is GG.In the second scenario, where the focus is on the probability that both children are boys ((BB)), given the older child is a boy, we only consider relevant combinations: BB and BG. Here, the single favorable outcome is BB.The ratio of favorable to possible outcomes determines our probability. For the older child as a boy leading to both boys, it's\(rac{1}{2}\).By singling out favorable outcomes, we succeed in deriving precise probabilities for each specified scenario.