Question

The sex ratio of newborn human infants is about 105 males : 100 females. \(^{21}\) If four infants are chosen at random, what is the probability that (a) two are male and two are female? (b) all four are male? (c) all four are the same sex?

Short answer

The probabilities are - (a) Two males and two females: \( \binom{4}{2} \left(\frac{105}{205}\right)^2 \left(\frac{100}{205}\right)^2 \), (b) All four males: \( \binom{4}{4} \left(\frac{105}{205}\right)^4 \), (c) All four the same sex: \( P(4, 0) + P(0, 4) \).

Step-by-step solution

Understanding the Problem

We are given the sex ratio of newborns and need to find probabilities for different compositions of sexes in a random group of four infants. The sex ratio suggests there are 105 males for every 100 females. We will calculate probabilities for three scenarios: two males and two females (a), all four being males (b), and all four infants having the same sex (c).

Calculating Total Probability for Males and Females

First, calculate the probability of selecting a male infant and a female infant. Since there are 105 males and 100 females, the total number of infants is 105 + 100 = 205. The probability of choosing a male is therefore 105/205 and for a female is 100/205.

Calculate a) Two Males and Two Females

We use the binomial probability formula for this scenario: \( P(X=k) = \binom{n}{k} p^k (1-p)^{(n-k)} \) where \( n = 4 \), \( k = 2 \), the probability of selecting a male \( p = \frac{105}{205} \) and \( 1-p = \frac{100}{205} \).Thus, the probability of getting exactly 2 males and 2 females is: \( P(2, 2) = \binom{4}{2} \left(\frac{105}{205}\right)^2 \left(\frac{100}{205}\right)^2 \).

Calculate b) All Four Are Male

Using the same formula with \( n = 4 \), \( k = 4 \), and \( p = \frac{105}{205} \), we get: \( P(4, 0) = \binom{4}{4} \left(\frac{105}{205}\right)^4 \left(\frac{100}{205}\right)^0 \).

Calculate c) All Four Are the Same Sex

The probability that all four are the same sex is the sum of the probabilities that all four are male or all four are female. We've already calculated the probability of all four being male, so we need to calculate the probability for all females and add it to the probability of all males. Probability for all females \( P(0, 4) \) is calculated similarly using the female probability: \( P(0, 4) = \binom{4}{0} \left(\frac{100}{205}\right)^4 \left(\frac{105}{205}\right)^0 \).Then sum \( P(4, 0) + P(0, 4) \) for the total probability of all four being the same sex.

Key concepts

Binomial Probability Formula

Grasping the concept of the binomial probability formula is essential for calculating the likelihood of an event with two possible outcomes. This formula is written as \( P(X=k) = \binom{n}{k} p^k (1-p)^{(n-k)} \), where \( P(X=k) \) represents the probability of \( k \) successes in \( n \) trials, \( p \) is the probability of success on a single trial, and \( (1-p) \) is the probability of failure.

When applying this to genetics, such as determining the probability of a sex distribution among newborns, remember that each birth is considered an independent trial with two outcomes: male or female. The binomial formula can help predict probabilities like how likely it is to have a certain number of male or female infants in a selected group. For example, in the exercise where we sought the probability of two males and two females out of four infants, we employed the binomial formula with the given success probabilities for males and females.

In more tangible terms, imagine you have a bag of 105 blue and 100 red marbles, each marble representing a male or female infant. If you draw a marble four times, replacing it each time to keep the ratio constant, the binomial probability formula would be used to calculate the exact likelihood of pulling a specific combination of blue and red marbles.

Sex Ratio

The sex ratio is a term used to describe the proportion of males to females in a population. In humans, at birth, it averages around 105 males to 100 females. This ratio can influence various genetic probability calculations, specifically when analyzing the expected distribution of sexes in a group of newborns. The sex ratio can be affected by numerous factors, including biological, environmental, and even social variables.

The ratio doesn't mean that every 205 infants will be composed of precisely 105 boys and 100 girls. But when assessing larger numbers, the sex ratio tends to reflect this proportion. To translate the sex ratio into probabilities required for the binomial formula, we express the likelihood of an infant being male or female based on this ratio. For instance, in our exercise with four infants, the probability of selecting a male can be calculated as the ratio of the number of males to the total population, which then becomes a key variable in our binomial probability formula.

Random Sampling

Random sampling refers to the process of selecting a subset of individuals from a population in such a way that every individual has an equal chance of being chosen. This method is vital in genetics and probability exercises to ensure that the samples are representative of the population, thus providing unbiased estimates of genetic ratios, like the sex ratio of newborns.

It's the foundation for exercises like ours, where we're choosing four infants 'at random'. The implication of this random selection is significant when applying the binomial probability formula. Assuming that each birth is an independent event and every child has an equal chance to be male or female, random sampling validates that the calculated probabilities based on the sex ratio will hold true across various random groups of newborns.

Moreover, when teaching the importance of random sampling, it's helpful to highlight that it helps in reducing the potential for systematic bias in the selection of samples and, therefore, in the predictions or conclusions that are drawn from those samples.