Problem 96 - Calculus for Biology and Medicine

Question

Use the Poisson approximation. For a certain vaccine, 1 in 500 individuals experiences some side effects. Find the probability that, in a group of 200 people, at least 1 person experiences side effects.

Step-by-Step Solution

Answer

The probability that at least one person experiences side effects is approximately 0.3297.

Step 1: Define the Problem

We need to find the probability that at least 1 person in a group of 200 experiences side effects. Each person independently experiences side effects with a probability of \( \frac{1}{500} \).

Step 2: Identify the Parameters for Poisson

To use Poisson approximation, calculate the expected number of people experiencing side effects (\lambda) by multiplying the group size by the individual probability: \( \lambda = 200 \times \frac{1}{500} \).

Step 3: Calculate \(\lambda\)

Plug the values into the formula: \( \lambda = 200 \times \frac{1}{500} = 0.4 \).

Step 4: Apply Poisson Probability Formula

The Poisson probability of exactly 0 people experiencing side effects is given by \( P(X = 0) = \frac{e^{-\lambda} \lambda^0}{0!} \).

Step 5: Compute \(P(X = 0)\)

Calculate using \( \lambda = 0.4 \): \( P(X = 0) = e^{-0.4} \approx 0.6703 \).

Step 6: Find the Probability of At Least One Person

Use the complement principle: \( P(X \geq 1) = 1 - P(X = 0) \), so \( P(X \geq 1) = 1 - 0.6703 \approx 0.3297 \).

Key Concepts

Probability Calculation

Probability calculation involves determining the chance of an event occurring. In our problem, we start with basic probability concepts. Consider that each person independently experiences side effects with a probability of \( \frac{1}{500} \). This is a very small chance, highlighting why individual probabilities may not be clear in larger contexts.

When dealing with multiple trials, like having 200 people, we are interested in a scenario probability. Is it more effective to use a simpler approximation method? In this instance, yes. The Poisson approximation becomes a powerful ally. With repeated and rare events, like vaccine side effects, calculating each event in detail might be cumbersome and unnecessary.

The small probability paired with many trials hints at a Poisson process. This allows us to substitute direct calculation with a more streamlined method, making probability calculations easier for specific cases. Remember:

• Identify if a method like Poisson can be used to simplify.
• Check the individual probability and trial number to see if it applies.

Expected Value

Expected value is a foundational concept in probability and statistics, kind of like the average outcome you would anticipate if you repeated an event many times.

In our problem, expected value is represented by \( \lambda \), the average number of people expected to have side effects in this scenario of 200 trials. We derive \( \lambda \) by multiplying the number of people by each individual’s chance of experiencing effects: \( \lambda = 200 \times \frac{1}{500} = 0.4 \).

This tells us that, on average, 0.4 people might be expected to experience side effects in this group. Knowing the expected value helps set up for further calculations, especially with Poisson distributions, where \( \lambda \) is integral to probability formulas.

The expected value is crucial when approximating outcomes, especially when actual events might not be practical to compute. Key points to keep in mind:

• Expected value provides an average basis for probability.
• Use expected value in setting up further probabilistic models.

Complement Rule

The complement rule simplifies complex probability questions. It's based on the principle that the probability of all possible outcomes in a scenario must add up to 1, or 100%.

In our problem, calculating the probability that at least one person experiences side effects directly seems tricky. However, if we restate this problem using the complement rule, it becomes simpler. Instead of calculating directly, determine the probability that no one experiences side effects \( P(X = 0) \) and subtract this from 1.

Using the Poisson approximation, we find \( P(X = 0) = e^{-0.4} \approx 0.6703 \). Therefore, \( P(X \geq 1) = 1 - P(X = 0) = 1 - 0.6703 \approx 0.3297 \). The complement rule allows us to elegantly solve problems like this without tackling the calculation head-on. Keeping in mind:

• Make complex calculations easier by finding complementary probabilities.
• The sum of probabilities for a full set of outcomes is always 1.
• Using complements is efficient in problems featuring multiple, cumulative events.