Problem 98 - Calculus for Biology and Medicine

Question

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.

Step-by-Step Solution

Answer

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

Step 1: 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., \(X = 0\)).

Step 2: 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.

Step 3: Compute the Exact Probability Using Binomial

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

Step 4: Simplify the Exact Probability

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

Step 5: 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 \( e^{-0.5} \approx 0.607 \).

Step 6: Compare and Conclude

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

Key Concepts

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.