Problem 33 - Calculus for Biology and Medicine

Question

A family has three children. Assuming a \(1: 1\) sex ratio, what is the probability that all of the children are girls?

Step-by-Step Solution

Answer

The probability that all three children are girls is \(\frac{1}{8}\).

Step 1: Understanding the problem

We need to find the probability that all three children in a family are girls. Since children can either be a girl or a boy, with equal probability, we will use the basic principles of probability.

Step 2: Determine the probability of a single event

The probability that one child is a girl is denoted by \(P(G) = \frac{1}{2}\). Since the sex ratio is 1:1, the probability of having a girl is equal to the probability of having a boy.

Step 3: Calculating the probability of multiple independent events

The probability of multiple independent events occurring together is the product of their probabilities. Here, we need all three children to be girls. The probability for each child is independent of the others, so we calculate: \[P(\text{3 Girls}) = P(G) \times P(G) \times P(G) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right)\]

Step 4: Simplifying the probability expression

Calculate the product: \[P(\text{3 Girls}) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}\]. Thus, the probability that all three children are girls is \(\frac{1}{8}\).

Key Concepts

Sex Ratio

A sex ratio is a concept in demography that refers to the relative number of males to females in a given population. It's commonly expressed as a ratio, such as 1:1, which signifies one male for every female. In our exercise, the sex ratio is 1:1, meaning the likelihood of any child being a boy or a girl is equal.

Understanding the sex ratio is essential when calculating probabilities concerning family compositions. In natural conditions, this ratio is approximately equal, which justifies using it in our probability calculations.

• Equal chance of being male or female: Each child has a probability of 0.5 to be a girl and 0.5 to be a boy, making the sex of each child equally probable.
• Applying the sex ratio: Knowing the ratio helps in setting the probability context in problems related to family and population sex distributions.

In reality, the global sex ratio at birth is slightly skewed due to biological and sociocultural factors, but for basic probability exercises, it's assumed to be 1:1.

Independent Events

In probability, events are considered independent if the occurrence of one does not influence the occurrence of another. This is a crucial aspect when it comes to calculating probabilities over multiple trials, such as determining the likelihood of all children in a family being of one gender.

For independent events:

• The outcome of one event does not affect the outcome of another.
• Probabilities can be multiplied to find the combined likelihood of two or more independent events occurring together.

In the exercise concerning the family of three children, each child's sex is an independent event. The birth of one girl does not influence the probability of a second or third child being a girl. Thus, to find the probability of having three girls, you multiply the probability for each child:\[P( ext{3 Girls}) = P(G) \times P(G) \times P(G)\]This independence simplifies many probability calculations and is a fundamental concept in mathematics.

Probability of a Single Event

The probability of a single event is a basic concept in statistics and probability theory that refers to the likelihood of one specific occurrence. An event, in this context, could be any outcome or set of outcomes, such as one child being a girl.

To find the probability of a single event, you consider the ratio of successful outcomes to the total number of possible outcomes. When discussing a single child being a girl, the probability \(P(G)\) is calculated as:
\[P(G) = \frac{\text{Number of ways to have a girl}}{\text{Total possible outcomes}}\]In our scenario:

• The probability of having either a girl or a boy is equal: each is \(\frac{1}{2}\).
• This forms the basis for calculating combined probabilities in sequences of independent events, such as siblings being of the same gender.

Understanding the probability of a single event helps to build the foundation for more complex calculations involving multiple events.