Problem 97 - Calculus for Biology and Medicine

Question

Use the Poisson approximation. 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 one 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 one Down syndrome case in 1000 births is about 80.2% using Poisson approximation, similar to the exact probability.

Step 1: Identify the problem type

We have a problem involving a rare event (Down syndrome) and a large sample size (1000 births). This scenario is suitable for a Poisson approximation.

Step 2: Define parameters for Poisson approximation

The probability of a birth being affected by Down syndrome is approximately 1/700. For 1000 births, the expected number of cases, \( \lambda \), is calculated as:\[\lambda = \frac{1000}{700} \approx 1.4286\].

Step 3: Poisson probability calculations

The Poisson probability function is defined as:\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]We compute the probabilities for at most one case, \( P(X \leq 1) \):\[P(X = 0) = \frac{e^{-1.4286} (1.4286)^0}{0!} \]\[P(X = 1) = \frac{e^{-1.4286} (1.4286)^1}{1!}\]Add these probabilities for \( X \leq 1 \):\[P(X \leq 1) = P(X = 0) + P(X = 1)\].

Step 4: Calculate the exact probability using Binomial distribution

For the binomial distribution, use \( n = 1000 \) and \( p = \frac{1}{700} \). Calculate the probability for \( X = 0 \) and \( X = 1 \) and sum them:\[P(X = 0) = \binom{1000}{0} (\frac{1}{700})^0 (\frac{699}{700})^{1000}\]\[P(X = 1) = \binom{1000}{1} (\frac{1}{700})^1 (\frac{699}{700})^{999}\]\[P(X \leq 1) = P(X = 0) + P(X = 1)\].

Step 5: Final calculations and comparisons

Calculate the numerical values from the Poisson and Binomial distributions. Compare the approximation to the exact probability to see how the Poisson approximation fares. Use a calculator or software for accurate computation of exponentials and factorials.

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials of a binary experiment. A binary experiment is one that has only two possible outcomes: success or failure. To model a situation using a binomial distribution, you typically need two parameters:

• n: The number of trials or experiments.
• p: The probability of success on each trial.

In the context of the original exercise, the number of trials, \( n \), is 1000, representing the 1000 births, and the probability of success, \( p \), is \( \frac{1}{700} \) as each birth has a 1 in 700 chance of being affected by Down syndrome.

When using a binomial distribution to determine the probability of at most one event occurring — such as having at most one case of Down syndrome among 1000 births — you compute the probability for each possible outcome (zero and one in this case) and sum them to find the cumulative probability:

• \( P(X = 0) \): Probability of zero cases
• \( P(X = 1) \): Probability of exactly one case

By adding these probabilities together, you get \( P(X \leq 1) \), which represents the probability of having at most one case.

Probability Calculations

Probability calculations involve working with numbers that express how likely an event is to occur on a scale from 0 to 1. A probability of 0 means the event is impossible, while a probability of 1 means it’s certain. For real-world applications, probabilities often indicate the expected chance of an event happening based on past data or theoretical assumptions. Standard probability calculations for different distributions rely on using the appropriate formulas and interpreting the results accurately:

• Poisson Probability — Used for rare events over a fixed interval, like time or space.
• Binomial Probability — Used for a fixed number of binary trials.

In practice, calculating probability involves several steps:

• Identify the type of distribution that best fits the scenario.
• Use the mathematical formula associated with the distribution.
• Perform calculations, often requiring factorial or exponential operations.

For precise calculations, especially in cases involving large numbers or small probabilities, calculators or computational software are commonly used. This approach minimizes errors and provides exact results.

Poisson Distribution

The Poisson distribution is a discrete probability distribution used to model the number of times an event occurs within a specific interval of time or space. It is particularly useful for modeling rare events, which happen relatively infrequently compared to the size of the potential sample space.To use a Poisson distribution, you need:

• \( \lambda \): The average number of occurrences in the interval.

This is calculated as \( \lambda = np \), where \( n \) is the number of trials, and \( p \) is the probability of success.

In the context of the original exercise, the calculation of \( \lambda = \frac{1000}{700} \approx 1.4286 \) takes into account the expected number of Down syndrome cases among 1000 births.The Poisson probability for an event's exact number, \( k \), occurring is given by:\[P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
For the scenario of at most one event, you calculate:

• \( P(X = 0) \): Probability of zero cases
• \( P(X = 1) \): Probability of exactly one case

Adding these probabilities helps estimate the likelihood of having one or fewer occurrences of a rare event, making the Poisson distribution a valuable approximation tool when the conditions of its use are met.