Problem 7 - Calculus for Biology and Medicine

Question

A family has two children. The younger one is a girl. Find the probability that the other child is a girl as well.

Step-by-Step Solution

Answer

The probability that both children are girls is \( \frac{1}{2} \).

Step 1: Determine Possible Outcomes

When determining outcomes for a family with two children, each child can either be a boy (B) or a girl (G). Given this, and knowing that the younger child is a girl, the possible combinations for the children's genders are (Older Child, Younger Child): (B, G) and (G, G).

Step 2: Identify the Event of Interest

We are interested in the event where both children are girls. From our possible outcomes, this event occurs if the combination is (G, G).

Step 3: Calculate the Probability of the Event

We know there are 2 possible outcomes: (B, G) and (G, G), and only 1 outcome that includes both children being girls (G, G). Since the outcomes are equally likely, the probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, which is \( \frac{1}{2} \).

Key Concepts

conditional probability

Conditional probability is a fascinating concept in probability theory. It involves calculating the likelihood of an event occurring based on a condition or criteria that is already known. To better understand it, think about our family scenario: we've been told that the younger child is a girl, and we’re asked to find the probability that the other child is also a girl.

When we know something ahead of time (like the condition that the younger child is a girl), it affects or "conditions" the probability of other outcomes.
This is different from basic probability because instead of considering all possible events, we narrow down the sample space. In this example, our sample space is limited to two possibilities:

• (B, G): the older is a boy, and the younger is a girl.
• (G, G): both are girls.

By looking at these possible outcomes, we notice the condition affects the probability calculation, since we ignore impossible scenarios like (G, B) and (B, B) due to the prior information. Calculating conditional probability involves counting the conditions that satisfy our event against those that occur under our given condition. Think of it like telling a story where you already know the ending to a part of it; you only need to figure out the rest.

combinatorics

Combinatorics is the mathematical study of counting combinations. It helps us understand how many different ways things can occur. In many probability problems, especially those involving arrangements or selections, combinatorics plays a central role.

In our example with the two children, combinatorics helps us list all the potential gender combinations based on the known condition. Despite there being two slots for each child that could be either a boy or a girl, since the younger child is already a girl, our combinations reduce:

• Older Child: Boy, Younger Child: Girl (B, G)
• Older Child: Girl, Younger Child: Girl (G, G)

By methodically listing combinations, we use combinatorics to simplify problem-solving. Understanding these combinations clarifies the sample space and the limits on possible outcomes when constraints (like one child’s gender) are in place. It allows us to solve the probability question efficiently by visualizing all potential scenarios clearly.

statistical outcomes

Statistical outcomes are the different results that can occur during an experiment or event, essential in probability calculations. Each result consists of a distinct possibility within the given context.
For the family gender scenario, statistical outcomes focus on the potential gender arrangements for the two children.

Once we establish the condition that the younger child is a girl, the outcomes are narrowed to two distinct possibilities: either the older child is a boy or a girl. This streanlining makes statistical outcomes a powerful tool; it shows clearly what results are possible under given conditions.
Recognizing statistical outcomes means identifying not just the outcomes but also understanding their probabilities. In our situation, the outcomes are equally likely, meaning each is as likely as the other. Therefore, calculating probabilities becomes a straightforward division of favorable cases by total cases. This process captures the core essence of calculating probability – translating known facts about scenarios into quantifiable results.