Problem 34 - Calculus for Biology and Medicine

Question

Turner's Syndrome Turner's syndrome is a chromosomal disorder in which girls have only one \(X\) chromosome. It affects about 1 in 2000 girls. About 1 in 10 girls with Turner's syndrome suffers from narrowing of the aorta. (a) In a group of 4000 girls, what is the probability that no girls are affected with Turner's syndrome? That one girl is affected? Two? At least three? (b) In a group of 170 girls affected with Turner's syndrome, what is the probability that at least 20 of them suffer from an abnormal narrowing of the aorta?

Step-by-Step Solution

Answer

For part (a): No girls: very small probability; One girl: higher but still small; Two girls: even smaller; At least three: small. For part (b): Probability of at least 20 girls with narrowing is higher, around 0.12 using normal approximation.

Step 1: Determine Probability of Turner's Syndrome

Given that Turner's syndrome affects 1 in 2000 girls, the probability that a single girl is affected, denoted as \( p \), is \( \frac{1}{2000} \). In contrast, the probability that a single girl is not affected is \( 1 - p = \frac{1999}{2000} \).

Step 2: Calculate Probability for No Girl with Turner's Syndrome

Using the binomial probability formula \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] for \( n = 4000\) and \(k = 0\), we calculate the probability of no girl being affected: \[ P(X = 0) = \binom{4000}{0} \left( \frac{1}{2000} \right)^0 \left( \frac{1999}{2000} \right)^{4000}. \]

Step 3: Calculate Probability for One Girl with Turner's Syndrome

Using the same formula for \( n = 4000\) and \( k = 1 \), we find the probability that exactly one girl is affected: \[ P(X = 1) = \binom{4000}{1} \left( \frac{1}{2000} \right)^1 \left( \frac{1999}{2000} \right)^{3999}. \]

Step 4: Calculate Probability for Two Girls with Turner's Syndrome

We now compute the probability for \( k = 2 \): \[ P(X = 2) = \binom{4000}{2} \left( \frac{1}{2000} \right)^2 \left( \frac{1999}{2000} \right)^{3998}. \]

Step 5: Determine Probability for At Least Three Girls with Turner's Syndrome

Since calculating each individual probability for \( k \geq 3 \) directly can be cumbersome, we use \[ P(X \geq 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2). \] Compute the earlier probabilities and subtract from 1 to find this probability.

Step 6: Probability of Aorta Narrowing in Turner’s Syndrome

With 170 girls affected and 1 in 10 suffering from narrowing of the aorta, we have \( p = \frac{1}{10} \). Use the normal approximation of binomial distribution to calculate the probability of at least 20 girls \[ P(Y \geq 20) = 1 - P(Y \leq 19). \]

Step 7: Approximate with Normal Distribution

For the normal approximation: calculate \( \mu = np = 170 \times \frac{1}{10} = 17 \) and \( \sigma^2 = np(1-p) = 170 \times \frac{1}{10} \times \frac{9}{10} = 15.3 \). Use the normal distribution \( N(17, 3.91)\) and find \( Z = \frac{19.5 - 17}{3.91} \) to get the probability using the Z-table.

Key Concepts

Turner's Syndrome

Turner's Syndrome is a unique chromosomal disorder that impacts around 1 in 2,000 female individuals. Girls with this condition lack a part or all of an X chromosome, which can affect their physical development, reproductive health, and other bodily functions.

In the realm of probability, understanding the likelihood of Turner's Syndrome occurring in a population helps in assessing risk factors and preparing adequate healthcare measures. When dealing with large groups, like 4,000 girls in a classroom, calculating the probability allows us to estimate how many might potentially be affected by Turner's Syndrome.

Using probability models gives a clearer picture of the expected number of cases. It's crucial for healthcare planning and understanding community health needs.

Binomial Distribution

The Binomial Distribution is a fundamental concept in statistics that describes the number of successes in a sequence of independent experiments. Consider each experiment as a binary outcome—success or failure.

In the context of Turner's Syndrome, each girl can either be affected or not affected by the condition. Given a large group size and a known probability of affection, we can make predictions about the number of girls likely affected.

• Formula: The probability of exactly \( k \) successes (in our case, girls affected by Turner's Syndrome) out of \( n \) trials (girls in the group) is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \] where \( p \) is the probability of being affected.
• This statistical modeling helps us calculate probabilities for various scenarios, like exactly one, two, or no girls being affected.

Normal Approximation

The Normal Approximation is a statistical technique used when dealing with large sample sizes in binomial distributions. It's an efficient way to handle complex calculations and estimate probabilities without directly computing them via the binomial formula.

When examining events like the narrowing of the aorta among those with Turner's Syndrome, the number of trials and the probability can create a cumbersome binomial problem.

• Normal approximation simplifies this by using a normal distribution as a go-to model, which is characterized by its mean and standard deviation.
• In practice, the mean \( \mu \) and variance \( \sigma^2 \) are used to approximate binomial results. For instance, the mean number of affected girls suffering from aorta narrowing can be given by \( \mu = np \), while the variance is \( \sigma^2 = np(1-p) \).
• This method provides an easier way to calculate probabilities by referring to standard statistical tables.

Harnessing the normal approximation is vital when dealing with large data sets to ensure accurate and efficient probability estimations.