Problem 98
Question
Use the Poisson approximation. 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.
Step-by-Step Solution
Verified Answer
Exact probability is approximately 0.6065 and Poisson approximation is about 0.6065.
1Step 1: Understanding the Problem
We want to find the probability that no boy in a group of 500 is affected by fragile X syndrome, knowing that 1 in 1000 boys is affected. We will first calculate the exact probability using the binomial distribution, then approximate it using the Poisson distribution.
2Step 2: Define the Binomial Distribution
The situation can be modelled with a binomial distribution where the number of boys, \( n = 500 \), and the probability of one boy being affected is \( p = \frac{1}{1000} \). The random variable \( X \) represents the number of affected boys.
3Step 3: Calculate Exact Probability Using Binomial Distribution
The formula for finding the exact probability is \( P(X = 0) = \binom{n}{0} p^0 (1-p)^n \). Here, \( 0 \) is the number of successes (affected boys), so \( P(X = 0) = (1-p)^n = (1 - \frac{1}{1000})^{500} \). Calculate this to find the exact probability.
4Step 4: Convert Problem for Poisson Approximation
The Poisson approximation is applicable when \( n \) is large and \( p \) is small, with \( \lambda = np \). Here, \( \lambda = 500 \times \frac{1}{1000} = 0.5 \). This \( \lambda \) can be used in the Poisson formula.
5Step 5: Calculate Probability Using Poisson Approximation
For a Poisson distribution, the formula is \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \), where \( k = 0 \). Thus, \( P(X = 0) = e^{-\lambda} = e^{-0.5} \). Compute this to find the Poisson approximate probability.
Key Concepts
Binomial distributionFragile X syndromeProbability calculation
Binomial distribution
The binomial distribution is a fundamental concept in probability theory, used to model the number of successes in a fixed number of independent experiments or trials. Each trial has two possible outcomes, often referred to as "success" and "failure." To use a binomial distribution, we need to know two key parameters: the number of trials, denoted as \( n \), and the probability of success in a single trial, \( p \).
For example, in the original problem concerning fragile X syndrome, the number of trials \( n \) is 500 (representing 500 boys), and the probability \( p \) of one boy having the syndrome is \( \frac{1}{1000} \). Using the binomial distribution, the probability of exactly \( k \) successes (i.e., boys affected) is given by the formula:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
In this problem, we wanted to find the probability that no boys were affected, so \( k = 0 \). This simplifies the formula to:
\[P(X = 0) = (1-p)^n = \left(1 - \frac{1}{1000}\right)^{500}\]
For example, in the original problem concerning fragile X syndrome, the number of trials \( n \) is 500 (representing 500 boys), and the probability \( p \) of one boy having the syndrome is \( \frac{1}{1000} \). Using the binomial distribution, the probability of exactly \( k \) successes (i.e., boys affected) is given by the formula:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
In this problem, we wanted to find the probability that no boys were affected, so \( k = 0 \). This simplifies the formula to:
\[P(X = 0) = (1-p)^n = \left(1 - \frac{1}{1000}\right)^{500}\]
Fragile X syndrome
Fragile X syndrome is a genetic condition that causes a range of developmental problems, including learning disabilities and cognitive impairment. It is one of the most common inherited causes of intellectual disabilities and also the most common known cause of autism or autism spectrum disorders. The condition affects both males and females, but males are more severely affected due to having only one X chromosome.
In statistical exercises such as the one presented, the syndrome typically appears infrequently in populations, which leads to using probability models to estimate how often it might occur in a given sample. Knowing that approximately 1 in 1000 boys is affected, helps us model this situation statistically using distributions like the binomial or Poisson, for estimating the relevancy of this rare occurrence in practice.
In statistical exercises such as the one presented, the syndrome typically appears infrequently in populations, which leads to using probability models to estimate how often it might occur in a given sample. Knowing that approximately 1 in 1000 boys is affected, helps us model this situation statistically using distributions like the binomial or Poisson, for estimating the relevancy of this rare occurrence in practice.
Probability calculation
Probability calculation is essential for understanding and predicting the likelihood of various outcomes in uncertain situations. There's a wide range of probability methods suitable for different types of data and situations.
When calculating probabilities using distributions:
In the fragile X syndrome example, after calculating the probability using the binomial distribution, a Poisson approximation provides a simpler computation under certain conditions. The Poisson formula for \( k \) events is:
\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
In our context, this means for \( k = 0 \), we use:
\[P(X = 0) = e^{-\lambda}\]
Understanding probability and these distributions allows us to make informed predictions about real-world events based on statistical models.
When calculating probabilities using distributions:
- **Binomial Distribution** is used when we have a fixed number of trials and each trial is independent with only two outcomes. It is perfect for predicting the number of times an event will occur in a series of experiments, like determining the probability of 0 boys being affected by fragile X syndrome in a group of 500.
- **Poisson Distribution** comes into play when the event is rare, and we want to approximate a binomial distribution when \( n \) is large and \( p \) is small. This is characterized through a parameter \( \lambda \), where \( \lambda = np \).
In the fragile X syndrome example, after calculating the probability using the binomial distribution, a Poisson approximation provides a simpler computation under certain conditions. The Poisson formula for \( k \) events is:
\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
In our context, this means for \( k = 0 \), we use:
\[P(X = 0) = e^{-\lambda}\]
Understanding probability and these distributions allows us to make informed predictions about real-world events based on statistical models.
Other exercises in this chapter
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