Problem 34 - Calculus for Biology and Medicine

Question

Turner's syndrome is a rare chromosomal disorder in which girls have only one \(X\) chromosome. The condition affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome suffers from an abnormal 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

About 32.33% for at least 3 girls in 4000, and about 7.17% for at least 20 with aortic narrowing among 170.

Step 1: Define the Probability and Random Variables

The probability that a girl is affected by Turner's syndrome in the United States is \( p = \frac{1}{2000} \). When examining a group of 4000 girls, we want to calculate the probability of 0, 1, 2, or at least 3 girls being affected using a binomial distribution: \( X \sim \text{Binomial}(n = 4000, p = \frac{1}{2000}) \).

Step 2: Calculate Binomial Probability for "0" Girls

Using the formula for binomial probability: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \), calculate \( P(X=0) \).\[P(X=0) = \binom{4000}{0} \left(\frac{1}{2000}\right)^0 \left(\frac{1999}{2000}\right)^{4000} \approx e^{-2} \approx 0.1353\]

Step 3: Calculate Binomial Probability for "1" Girl

Similarly, calculate the probability for one girl being affected:\[P(X=1) = \binom{4000}{1} \left(\frac{1}{2000}\right)^1 \left(\frac{1999}{2000}\right)^{3999} \approx 0.2707\]

Step 4: Calculate Binomial Probability for "2" Girls

Now calculate the same for two girls:\[P(X=2) = \binom{4000}{2} \left(\frac{1}{2000}\right)^2 \left(\frac{1999}{2000}\right)^{3998} \approx 0.2707\]

Step 5: Calculate Probability for at Least "3" Girls

To find the probability that at least 3 girls are affected, use the complement rule:\[P(X\geq3) = 1 - (P(X=0) + P(X=1) + P(X=2)) \approx 0.3233\]

Step 6: Define Probability for Aortic Narrowing

For the second part, given that 1 in 10 girls with Turner's syndrome have aortic narrowing, the binomial distribution is \( Y \sim \text{Binomial}(n = 170, p = 0.1) \). Calculate the probability of at least 20 girls with narrowing.

Step 7: Calculate Probability for at Least "20" Girls With Aortic Narrowing

Using the binomial cumulative formula, calculate:\[P(Y\geq 20) = 1 - P(Y < 20)\]Use a binomial table, calculator, or approximation to find \( P(Y < 20) \). After calculations:\[P(Y\geq20) \approx 0.0717\]

Step 8: Conclusion

In a group of 4000 girls, there's a 32.33% chance at least 3 have Turner's syndrome. In 170 girls with the syndrome, there's a 7.17% chance at least 20 have aortic narrowing.

Key Concepts

Turner's Syndrome

Turner's syndrome is a genetic condition specifically affecting girls and women. This condition occurs when one of the two X chromosomes in females is missing or significantly altered. Normally, females have two X chromosomes, but those with Turner's syndrome have only one, which leads to various developmental and medical challenges.

Some of the common features of Turner's syndrome include short stature, ovarian insufficiency, and certain heart defects such as aortic narrowing. The condition is relatively rare, affecting about 1 in every 2000 female births globally.

Although Turner's syndrome can lead to health issues, early diagnosis and tailored medical care can help individuals manage the condition effectively. Regular health monitoring and specific treatments can address many of the symptoms and improve quality of life.

Binomial Distribution

In probability theory, the binomial distribution is used to model the number of successful outcomes in a fixed number of trials, where each trial has the same probability of a particular outcome. A classic use is predicting events with two possible outcomes, like turning a coin heads or tails.

For instance, in the context of Turner's syndrome in a large group of girls, each individual girl represents a 'trial.' Here, the 'success' might mean a girl having Turner's syndrome, with the probability of success given as 1 in 2000. The formula for binomial probability is:

• For determining the probability of getting exactly k successes (e.g., 0, 1, or 2 girls affected with Turner's syndrome) in n independent trials, use:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

A binomial distribution is helpful for solving problems that involve probability in situations where there are a large number of trials and a small probability of success on each trial.

Probability Calculation

Probability calculations are essential for assessing the likelihood of specific outcomes in uncertain situations. By leveraging mathematical formulas, one can predict how likely an event is to occur given certain conditions.

To illustrate, consider a large group of 4000 girls. We want to calculate the probabilities of various events concerning Turner's syndrome using the binomial distribution.

• To find the likelihood of no girls having the syndrome:\[ P(X=0) = \binom{4000}{0} \left(\frac{1}{2000}\right)^0 \left(\frac{1999}{2000}\right)^{4000} \approx 0.1353 \]
• Similarly, for one or two girls affected:\[ P(X=1) \approx 0.2707 \] \[ P(X=2) \approx 0.2707 \]
• To determine the probability of at least three girls having Turner's syndrome, use the complement rule:\[ P(X \geq 3) = 1 - (P(X=0) + P(X=1) + P(X=2)) \approx 0.3233 \]

By using these calculations, students can not only solve statistical problems but also gain insight into how mathematical models reflect real-world phenomena.