Problem 88 - Calculus for Biology and Medicine

Question

Suppose the number of typos on a book page is Poisson distributed with mean \(0.1\). (a) Find the probability that there are no typos on a page. (b) How many pages with typos do you expect in a 200 -page book?

Step-by-Step Solution

Answer

(a) The probability of no typos is approximately 0.9048. (b) Expect about 19 pages with typos.

Step 1: Understanding the Poisson Distribution

The exercise mentions that the number of typos follows a Poisson distribution with a mean (\(\lambda\)) of 0.1. The Poisson probability mass function is given by \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \) where \( k \) is the number of events.

Step 1: Calculate Probability of No Typos

To find the probability of zero typos (\( k = 0 \)), use the Poisson formula: \( P(X=0) = \frac{e^{-0.1} 0.1^0}{0!} = e^{-0.1} \times 1 = e^{-0.1} \). Calculate: \( e^{-0.1} \approx 0.9048 \). Thus, the probability of no typos on a single page is approximately 0.9048.

Step 2: Calculate Expected Pages with Typos

For part (b), we want the expected number of pages with typos in 200 pages. The expected number of pages with at least one typo is determined by the probability of a typo on a single page (1 - probability of no typos), multiplied by the total number of pages.The probability of having at least one typo on a page is \(1 - P(X=0) = 1 - 0.9048 = 0.0952\). Thus, the expected number of pages with typos is \(200 \times 0.0952 = 19.04\).

Key Concepts

Mean (Lambda)

In a Poisson distribution, the mean, commonly denoted by \( \lambda \), plays a crucial role in defining the distribution. It represents the average rate or count of events happening in a fixed interval of time or space. In our example of typos on a book page, the mean \( \lambda = 0.1 \) indicates that on average, there are 0.1 typos per page.
This essentially means that over many pages, we'd expect 10 typos to occur for every 100 pages. Since \( \lambda \) is a small number here, it suggests that typos are rare. Understanding \( \lambda \) helps us grasp the behavior of the random variable we are dealing with.
The mean \( \lambda \) is not only the average rate but also the variance of the Poisson distribution. So, it provides a measure of dispersion, indicating how much variation or spread exists within the distribution. Knowing this is vital for solving problems involving predictable patterns of rare events.

Probability Mass Function

The probability mass function (PMF) of the Poisson distribution is the formula that gives us the probability of a certain number of events, \( k \), occurring. It is mathematically expressed as:

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

Here, \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and \( k! \) denotes the factorial of \( k \), meaning the product of all positive integers up to \( k \).
In the context of the exercise, we use this function to determine the probability of finding zero typos on a page. Substituting \( \lambda = 0.1 \) and \( k = 0 \), we calculate:
\[ P(X=0) = \frac{e^{-0.1} \times 0.1^0}{0!} = e^{-0.1} \]This simplifies to \( 0.9048 \), representing the probability that a single page will have no typos. The PMF thus serves as a tool to evaluate the likelihood of different outcomes within a Poisson process.

Expected Value

The expected value in probability and statistics refers to the long-run average or mean of a random variable, providing a measure of the central tendency of the distribution. In the context of this problem, we wish to find out how many pages with typos we expect in a 200-page book.
First, we determine the probability of at least one typo per page. Since no typo probability is 0.9048, we subtract this from 1:
\[ 1 - 0.9048 = 0.0952 \]
This means there is a 9.52% chance of having a typo on any given page.
The expected number of pages with one or more typos over 200 pages is calculated by multiplying this probability by the total number of pages:
\[ 200 \times 0.0952 = 19.04 \]
This gives us an expected value of 19.04 pages where we would find typos. Expected value is a vital concept as it allows us to anticipate average outcomes over multiple trials or cases.