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Chapter 12: Problem 98

Use the Poisson approximation. Fragile \(X\) Syndrome About 1 in 1000 boys is affected by fragile \(X\) syndrome, a genetic disorder that causes learning difficulties. Find the probability that, in a group of 500 boys, nobody is affected by this disorder by (a) computing the exact probability and (b) using a Poisson approximation.

Short Answer

Expert verified
The probability is approximately 0.606 with binomial, and 0.607 using Poisson approximation.

Step by step solution

01

Define the Parameters

The probability of a boy being affected by fragile \(X\) syndrome is \(p = \frac{1}{1000} = 0.001\). We have \(n = 500\) boys, meaning the number of trials is 500. We want to find the probability that no one is affected (i.e., \(X = 0\)).
02

Compute the Expected Number of Events

Calculate the expected number of boys affected by the syndrome using the formula for expected value: \(\lambda = n \times p = 500 \times 0.001 = 0.5\). This \(\lambda\) will be used in the Poisson approximation.
03

Compute the Exact Probability Using Binomial

The exact probability can be calculated using the binomial distribution formula. \( P(X = 0) = \binom{500}{0} (0.001)^0 (0.999)^{500} = 1 \times 1 \times (0.999)^{500} \).
04

Simplify the Exact Probability

Calculate \((0.999)^{500}\). Using a calculator, \((0.999)^{500} \approx 0.606\). Hence, the exact probability \(P(X = 0) \approx 0.606\).
05

Use Poisson Approximation

For \(X\) following a Poisson distribution with \(\lambda = 0.5\), the probability is given by \(P(X = 0) = \frac{e^{-\lambda} \lambda^0}{0!} = e^{-0.5} \approx 0.607\). This uses the fact that \( e^{-0.5} \approx 0.607 \).
06

Compare and Conclude

The exact probability of 0.606 is very close to the Poisson approximation of 0.607, verifying the approximation's validity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fragile X Syndrome
Fragile X syndrome is a genetic disorder that impacts cognitive development, leading to learning challenges. It is caused by a mutation on the X chromosome, which affects the production of a protein crucial for brain development. While it can occur in both boys and girls, it is more commonly diagnosed in boys, with an occurrence rate of about 1 in 1000 male births.

This disorder often results in intellectual disability, behavioral characteristics, and various physical features specific to Fragile X syndrome. Supportive interventions, educational programs, and genetic counseling can play significant roles in managing the condition and supporting the affected individuals and their families.
Binomial Distribution
The binomial distribution is used to model the number of successes in a fixed number of trials of a binary experiment, which means each trial has two possible outcomes (success or failure). It is defined by two parameters: the number of trials, denoted as 'n', and the probability of success on a single trial, denoted as 'p'.

To calculate the probability of exactly 'k' successes, the binomial formula \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]is applied. In this exercise, we have a scenario with 500 trials, each representing one boy being tested for Fragile X syndrome, and the probability of a boy being affected is 0.001.
Probability Calculation
Probability calculation in this context involves computing the likelihood that no boys in our sample group are affected by Fragile X syndrome. Using the binomial distribution equation, we calculate the exact probability that none of the 500 boys is affected by setting 'X' to zero.

This involves substituting the given parameters into the binomial formula and simplifying. For this case, it results in a probability of about 0.606, meaning there is a 60.6% chance that none of the boys in the group of 500 are affected.
Expected Value
The expected value in probability provides a measure of the center of a probability distribution, often representing the long-run average outcome of a random variable after many trials. It is calculated by multiplying the total number of trials by the probability of success for each trial.

In our scenario, the expected number of boys affected by Fragile X syndrome in a group of 500 is computed as \(\lambda = n \times p = 500 \times 0.001 = 0.5\). This expected value serves as the parameter for the Poisson distribution, illustrating a practical approximation under certain conditions when the number of trials is large, and the probability of success is small.

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Most popular questions from this chapter

Suppose the number of phone calls arriving at a switchboard per hour is Poisson distributed with mean 3 calls per hour. (a) Find the probability that at least one phone call arrives between noon and \(1 \mathrm{P.M}\). (b) Assuming that phone calls in different hours are independent of each other, find the probability that no phone calls arrive between noon and 2 P.M.

Let \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\) denote a sample of size \(n\). Show that $$ \sum_{k=1}^{n}\left(X_{k}-\bar{X}\right)=0 $$ where \(\bar{X}\) is the sample mean.

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5 .\) Use a graphing calculator to numerically approximate the median lifetime if the hazard-rate function is $$ \lambda(x)=0.5+0.1 e^{0.2 x}, \quad x \geq 0 $$

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5 .\) If the life span of an organism is exponentially distributed, and if \(x_{m}=4\) years, what is the hazard-rate function?

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?
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