Using Chebyshev and other inequality formulas (maybe even Central Limit Theorem)

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SUMMARY

This discussion focuses on calculating probabilities related to Turner's syndrome using statistical methods. Specifically, it addresses two problems: the probability of 0, 1, 2, or at least 3 girls affected by Turner's syndrome in a group of 4000, and the probability of at least 20 girls with Turner's syndrome suffering from an abnormal narrowing of the aorta in a group of 170. The recommended statistical approaches include the binomial distribution for both problems, with a Poisson distribution suggested for the first due to the low probability of occurrence.

PREREQUISITES
  • Understanding of binomial distribution
  • Familiarity with Poisson distribution
  • Knowledge of normal approximation techniques
  • Basic concepts of probability theory
NEXT STEPS
  • Study the properties and applications of the Poisson distribution
  • Learn about the binomial distribution and its calculations
  • Explore normal approximation methods in statistics
  • Investigate the Central Limit Theorem and its implications
USEFUL FOR

Statisticians, data analysts, and students studying probability theory, particularly those interested in applying statistical methods to medical conditions like Turner's syndrome.

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Turner's syndrome is a rare chromosomal disorder in which girls have only one X chromosome. It affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome also suffer from an abnormal narrowing of the aorta.

a. In a group of 4000 girls, what is the probability that there are 0,1,2, or at least 3 girls affected with Turner's syndrome?

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?
 
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For both questions the most direct approach would be using the binomial distribution, although for (b.) a normal approximation would make calculation easier.
 
For a., a Poisson distribution is also appropriate (and probably easier to work with than the Binomial), since the probability of the disorder is so low.