Problem 29 - Calculus for Biology and Medicine

Question

For \(n=50\) and \(p=0.1\), compute \(P\left(S_{n}=5\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Step-by-Step Solution

Answer

The exact value is approximately 0.174. Poisson gives 0.175, while normal approximation provides about 0.171.

Step 1: Define the Problem

We are given a binomial distribution with parameters \(n = 50\) and \(p = 0.1\). We need to find \(P\left(S_{n}=5\right)\), where \(S_n\) represents the number of successes in \(n\) trials.

Step 2: Exact Solution Using Binomial Formula

Use the binomial probability formula: \( P(S_n = k) = \binom{n}{k} p^k (1-p)^{n-k} \). For \(k = 5\), plug in the values: \( \binom{50}{5} (0.1)^5 (0.9)^{45} \). Using a calculator or computational tool for binomial calculations, evaluate this expression to find the exact probability.

Step 3: Poisson Approximation

Since \(n = 50\) is large and \(p = 0.1\) is small, the Poisson distribution can approximate the binomial. Calculate the parameter \(\lambda = np = 50 \times 0.1 = 5\). Then use \(P(X = 5) = \frac{e^{-\lambda} \lambda^5}{5!}\). Calculate this using \(\lambda = 5\) and find the probability.

Step 4: Normal Approximation

For a normal approximation, use the mean \(np = 5\) and the standard deviation \(\sqrt{np(1-p)} = \sqrt{50 \times 0.1 \times 0.9}\). Normalize \(k = 5\) using the continuity correction: calculate \(z = \frac{5.5 - 5}{\sqrt{np(1-p)}}\). Use the standard normal distribution (Z-table) to find \(P(4.5 < S_n < 5.5)\).

Step 5: Conclusion

Compare all the results obtained from the exact binomial, Poisson, and normal approximations to verify and understand the differences and accuracy of each method.

Key Concepts

Poisson Approximation

The Poisson approximation is a handy tool when dealing with binomial distributions, especially when the number of trials, denoted as \( n \), is large, and the success probability, \( p \), is small. This approximation simplifies complex calculations by converting the binomial distribution into a more manageable form, the Poisson distribution.

• The key parameter for the Poisson distribution is \( \lambda \), which is the product of \( n \) and \( p \), i.e., \( \lambda = np \).
• When using the Poisson distribution, the probability mass function is given by \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \), where \( k \) is the number of successes we want to find the probability for.

In the given problem, since \( n = 50 \) and \( p = 0.1 \), we have \( \lambda = 5 \). This allows us to calculate the probability of observing exactly five successes using a much simpler formula than if we had stuck with the complex binomial calculations. The beauty of the Poisson approximation lies in its efficiency and ease of use in applicable scenarios.

Normal Approximation

A normal approximation is another method used to approximate a binomial distribution. It becomes particularly useful when \( n \) is large, even without a small \( p \). This method leverages the Central Limit Theorem, which suggests that the distribution of sample means will be normal if the sample size is large enough.

• The approximate normal distribution has a mean \( \mu = np \), and its standard deviation is \( \sigma = \sqrt{np(1-p)} \).
• To improve accuracy, we use a continuity correction by adjusting \( k \) by 0.5. For example, instead of calculating \( P(S_n = 5) \), we compute \( P(4.5 < S_n < 5.5) \).
• With this correction, we then convert our range into the standard normal distribution using a \( z \)-score, which is calculated by \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the boundary of our range.

In our case, with \( n = 50 \) and \( p = 0.1 \), we calculated the mean to be 5 and the standard deviation as \( \sqrt{4.5} \). Applying the continuity correction and finding \( z \) allows us to use \( z \)-tables or calculators to determine the probability effectively.

Probability Calculations

In probability theory, calculating probabilities is crucial, especially when dealing with distributions. These calculations help us predict the likelihood of various outcomes and make informed decisions.

• For a binomial distribution, the probability of obtaining exactly \( k \) successes in \( n \) trials is determined using the formula: \( P(S_n = k) = \binom{n}{k} p^k (1-p)^{n-k} \).
• This formula requires evaluating combinations, \( \binom{n}{k} \), which counts how many ways we can choose \( k \) successes out of \( n \) trials. The terms \( p^k \) and \( (1-p)^{n-k} \) represent the probability of those successes and the complementary failures respectively.
• Exact calculations can be time-consuming and complex, especially with large \( n \). Hence, approximations like the Poisson and normal are often used to simplify the process.

Probability calculations form the backbone of many statistical methods, providing a foundation upon which more advanced models and analyses are based. Whether students are using precise formulas or helpful approximations, understanding these calculations equips them to tackle a wide range of problems in probability.