Problem 95 - Calculus for Biology and Medicine

Question

Use the Poisson approximation. For a certain vaccine, 1 in 1000 individuals experiences some side effects. Find the probability that, in a group of 500 people, nobody experiences side effects.

Step-by-Step Solution

Answer

The probability that nobody experiences side effects is approximately 0.6065.

Step 1: Understand the Poisson Approximation

The Poisson approximation is useful when a binomial probability distribution has a large number of trials, and the probability of success is small. In this problem, the probability of experiencing side effects (success) is small \( p = \frac{1}{1000} \) and the number of trials \( n = 500 \) is large. Therefore, it's appropriate to use a Poisson distribution to approximate this.

Step 2: Calculate Lambda (\(\lambda\))

The parameter \( \lambda \) for the Poisson distribution is calculated as \( \lambda = n \times p \). Substituting the values for \( n = 500 \) and \( p = \frac{1}{1000} \), we find \( \lambda = 500 \times \frac{1}{1000} = 0.5 \). This is the expected number of individuals experiencing side effects in a group of 500 people.

Step 3: Use the Poisson Probability Formula

The Poisson probability mass function is given by \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where \( k \) is the number of successes we are finding the probability for. In this case, we want to find the probability that \( k = 0 \), meaning no one experiences side effects: \[ P(X = 0) = \frac{e^{-0.5} \cdot 0.5^0}{0!} \] Simplifying this gives:\[ P(X = 0) = e^{-0.5} = \frac{1}{e^{0.5}} \]

Step 4: Calculate the Probability

Now calculate \( e^{-0.5} \). Using a calculator, you find that \( e^{-0.5} \approx 0.6065 \). Thus, the probability that nobody experiences side effects in a group of 500 people is approximately 0.6065.

Key Concepts

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In simple terms, it tells us how likely different events are to happen. There are several types of probability distributions, with the Poisson distribution being one of them.

The Poisson distribution is particularly useful when we expect a small number of events to happen within a fixed period or in a small area, and these events happen independently of each other. It is often used as an approximation when we have a large number of trials, such as in this exercise, where we have 500 people in the group. This distribution helps in figuring out the probability of a certain number of events happening, like the occurrence of side effects from a vaccine.

Parameter Lambda

Understanding the parameter lambda (\( \lambda \)) is crucial when dealing with a Poisson distribution. Lambda is the parameter that represents the average number of events occurring in a given time period or space. In simpler words, it's the expected number of occurrences.

To calculate lambda, you multiply the number of trials by the probability of a single success. For this exercise, the probability of experiencing a side effect after a vaccination is 0.001 (or 1 in 1000), and the number of individuals is 500, which makes lambda 0.5.

• \( n = 500 \) (number of trials)
• \( p = \frac{1}{1000} = 0.001 \) (probability of success)
• \( \lambda = n \times p = 0.5 \)

This lambda figure of 0.5 indicates that, on average, 0.5 individuals out of 500 would experience side effects.

Poisson Probability Formula

The Poisson probability formula helps us to calculate the likelihood of a given number of events occurring in a fixed interval. The formula is written as:

\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]

In this formula:

• \( e \) is a constant, approximately equal to 2.71828
• \( \lambda \) is the expected number of occurrences
• \( k \) is the number of occurrences for which we are finding the probability
• \( k! \) is the factorial of \( k \), or \( k \times (k - 1) \times (k - 2) \times \ldots \times 1 \)

In our example exercise, we are calculating the probability for \( k = 0 \) (i.e., nobody experiences side effects), and \( \lambda = 0.5 \). Substituting these values into the formula gives us: \[ P(X = 0) = \frac{e^{-0.5} \times 0.5^0}{0!} \] The result is approximately 0.6065, meaning there is a 60.65% chance that no one in the group of 500 people will experience side effects. This application of the formula showcases the practical use of Poisson probability.