Problem 31 - Calculus for Biology and Medicine

Question

Assume a \(1: 1\) sex ratio. A family has three children. Find the probability of each event: (a) \(A=\\{\) all children are girls \(\\}\) (b) \(B=\\{\) at least one boy \(\\}\) (c) \(C=\\{\) at least two girls \(\\}\) (d) \(D=\\{\) at most two boys \(\\}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{8}\); (b) \(\frac{7}{8}\); (c) \(\frac{1}{2}\); (d) \(\frac{7}{8}\).

Step 1: Understand the Problem

We need to calculate probabilities related to the gender of three children, treating the probability of a child being a girl or boy as equal at 1/2. We have events involving combinations of boys and girls in three births.

Step 2: Calculate the Total Possible Outcomes

Each child can either be a boy or a girl. Thus, for three children, there are a total of \(2^3 = 8\) possible combinations: (Boy, Boy, Boy), (Boy, Boy, Girl), (Boy, Girl, Boy), (Boy, Girl, Girl), (Girl, Boy, Boy), (Girl, Boy, Girl), (Girl, Girl, Boy), (Girl, Girl, Girl).

Step 3: Event A - All Children are Girls

The event where all three children are girls correspond only to one outcome: (Girl, Girl, Girl). Thus, the probability of event A (all girls) is \(\frac{1}{8}\).

Step 4: Event B - At Least One Boy

The complement of event B (at least one boy) is having no boys, i.e., all children are girls. Using the complement rule, probability of at least one boy \( = 1 - \text{Probability of all girls} = 1 - \frac{1}{8} = \frac{7}{8}\).

Step 5: Event C - At Least Two Girls

To find the probability of at least two girls, we consider the outcomes (Girl, Girl, Girl), (Girl, Girl, Boy), and (Girl, Boy, Girl), (Boy, Girl, Girl). There are 4 favorable outcomes, so the probability is \(\frac{4}{8} = \frac{1}{2}\).

Step 6: Event D - At Most Two Boys

This event excludes (Boy, Boy, Boy) and includes the remaining 7 combinations. Probability of at most two boys is \(\frac{7}{8}\).

Key Concepts

Sex Ratio

In genetics, the concept of sex ratio refers to the proportion of males to females in a population. A typical default assumption in many genetic probability problems is a 1:1 sex ratio. This means that each child born has an equal chance of being a girl or a boy.

• For example, if we assume a 1:1 sex ratio, then the probability that a baby is a girl is 0.5 (or 50%), and similarly, the probability that a baby is a boy is also 0.5.
• This assumption makes it possible to calculate probabilities of various combinations of boys and girls in a family.

In probability theory, when considering a situation with three children, we count the possible outcomes. With a 1:1 sex ratio, each child is like flipping a fair coin, where each outcome (boy or girl) is equally likely. Therefore, for three children, there are 2 x 2 x 2 = 8 possible gender combinations.

Probability Calculation

Probability calculation involves determining how likely a certain event is to occur among all possible outcomes. In the context of our genetics example with three children, it involves calculating how often we expect certain combinations of boys and girls to occur.

• First, we outline all possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.
• Each of these outcomes is equally likely given a 1:1 sex ratio and each has a probability of \(\frac{1}{8}\) of occurring.

Next, we determine the specific event probabilities:

• Event A, where all children are girls, refers only to the 'GGG' outcome. The probability is \(\frac{1}{8}\).
• Event B, where at least one boy is present, includes everything except 'GGG', yielding a probability of \(1 - \frac{1}{8} = \frac{7}{8}\).
• Event C, involving at least two girls, includes the outcomes GGG, GGB, GBG, and BGG, resulting in a probability of \(\frac{4}{8} = \frac{1}{2}\).
• Event D, having at most two boys, encompasses all except 'BBB', with a probability of \(\frac{7}{8}\).

Genetics Probability Problems

Problems dealing with genetics probability often require understanding the fundamental principles of probability alongside biological concepts like sex determination. They are key in assessing risks and predictions in genetics, especially when dealing with inheritance patterns within families.

• For instance, predicting the probability of having boys or girls can inform understanding of potential genetic disorders more likely to occur in one sex.
• Such calculations assume independent events, where the sex of one child does not affect the sex of another.

These problems require the use of probability theories and rules, like the complement rule, to determine probabilities:

• The complement of getting all girls, \(\frac{1}{8}\), is used to calculate the probability of having at least one boy, \(1 - \frac{1}{8}\).

Thus, understanding these foundational concepts is critical for solving genetic probability challenges efficiently.