Problem 97 - Calculus for Biology and Medicine

Question

Use the Poisson approximation. Down Syndrome About 1 in 700 births in the United States is affected by Down syndrome, a chromosomal disorder. Find the probability that there is at most 1 case of Down syndrome among 1000 births by (a) computing the exact probability and (b) using a Poisson approximation.

Step-by-Step Solution

Answer

The probability of at most 1 case is approximately the same for both the exact binomial and Poisson approximation.

Step 1: Define the Problem Using Binomial Distribution

We are given that 1 in 700 births is affected by Down syndrome. So, the probability \( p \) that any single birth is affected is \( \frac{1}{700} \). We are considering 1000 births, which means \( n = 1000 \). Therefore, we have a binomial distribution with parameters \( n = 1000 \) and \( p = \frac{1}{700} \).

Step 2: Calculate the Binomial Probability for At Most 1 Case

First, calculate the probability of exactly 0 cases using the binomial formula: \[ P(X=0) = \binom{1000}{0} \left( \frac{1}{700} \right)^0 \left( 1 - \frac{1}{700} \right)^{1000} = \left( \frac{699}{700} \right)^{1000}. \]Then, calculate the probability of exactly 1 case:\[ P(X=1) = \binom{1000}{1} \left( \frac{1}{700} \right)^1 \left( 1 - \frac{1}{700} \right)^{999} = 1000 \times \frac{1}{700} \times \left( \frac{699}{700} \right)^{999}. \]Sum these probabilities to find the probability of at most 1 case: \[ P(X \leq 1) = P(X=0) + P(X=1). \]

Step 3: Introduce Poisson Approximation

Since the probability of each birth having Down syndrome is small \( p = \frac{1}{700} \) and the number of trials is large \( n = 1000 \), we can use the Poisson approximation for the binomial distribution. The Poisson parameter \( \lambda \) is given by \( \lambda = n \times p \). So, \( \lambda = 1000 \times \frac{1}{700} \approx 1.4286 \).

Step 4: Calculate Poisson Probability for At Most 1 Case

Use the Poisson probability formula:\[ P(Y = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \]where \( k \) is the number of cases. Calculate for 0 and 1 cases:\[ P(Y = 0) = \frac{e^{-1.4286} (1.4286)^0}{0!} = e^{-1.4286}, \]\[ P(Y = 1) = \frac{e^{-1.4286} (1.4286)^1}{1!} = 1.4286 \times e^{-1.4286}. \]Sum these probabilities to get the probability of at most 1 case:\[ P(Y \leq 1) = P(Y=0) + P(Y=1). \]

Step 5: Compare the Results

Compare the exact probability \( P(X \leq 1) \) from the binomial calculation and the approximate probability \( P(Y \leq 1) \) from the Poisson. Note that the Poisson approximation is typically very close to the exact binomial solution when \( n \) is large and \( p \) is small.

Key Concepts

Understanding Binomial Distribution

The binomial distribution is quite intuitive once you break it down. It models the probability of achieving a given number of successes in a fixed number of trials. In our Down syndrome example, each birth is considered a trial, and a birth being affected by Down syndrome is defined as a 'success'.
In a binomial distribution, two parameters are crucial: the number of trials, \( n \), and the probability of success on each trial, \( p \). In our case, we have 1000 births (or trials) with a success probability \( p = \frac{1}{700} \). Each birth is independent, meaning the outcome of one birth doesn't affect another.
To calculate the probability of at most one case of Down syndrome, we need the sum of the probabilities of exactly zero and exactly one case. This mirrors our need to understand both the specific outcome probabilities and the combined outcomes of interest.

Breaking Down Probability Calculation

Probability calculations can often be broken into simpler parts, especially with distributions like the binomial and Poisson. In our exercise, we calculated the probability for 0 and 1 birth being affected and summed these probabilities to find the overall probability of at most 1 case.
The calculation of probability using the binomial distribution for 0 cases uses the formula: \[P(X=0) = \binom{1000}{0} \left( \frac{1}{700} \right)^0 \left( 1 - \frac{1}{700} \right)^{1000} = \left( \frac{699}{700} \right)^{1000}.\] To expand, with exactly one affected birth:\[P(X=1) = \binom{1000}{1} \left( \frac{1}{700} \right)^1 \left( 1 - \frac{1}{700} \right)^{999}.\]
By summing these, \[P(X \leq 1) = P(X=0) + P(X=1),\] we determine the probability of such events collectively, providing insight into this specific population model.

Chromosomal Disorders Like Down Syndrome

Chromosomal disorders are conditions arising from abnormalities in chromosome number or structure. These conditions can lead to various physical and intellectual challenges. Down syndrome, the focus of our exercise, is a common chromosomal disorder caused by an extra copy of chromosome 21.
It's a condition that occurs in about 1 in 700 births, and probability tools like the binomial and Poisson distributions help to model and predict its occurrence in larger populations. This knowledge not only aids medical professionals but also informs public health strategies and prepares systems to support affected individuals.

• This systematic approach helps predict occurrences in large batches, like 1000 births, and manage potential healthcare needs.
• Understanding probabilities offers a powerful tool to translate statistical data into actionable health policies.

Probability models guide us to use large datasets efficiently to deduce outcomes, crucial in the strategic planning of healthcare provisions.