Problem 41 - A First Course in Probability

Question

A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the \(20 ?\) What assumptions are you making?

Step-by-Step Solution

Answer

In conclusion, when catching 20 fish from a pond with 100 fish out of which 30 are carp, we can model this with a hypergeometric distribution having a mean number of carps of 6 and a variance of approximately 2.8282, under the assumption that each fish has an equal chance of being caught independently.

Step 1: Identify the hypergeometric distribution

The catching of fishes can be modeled as a hypergeometric distribution, where we have a population of size \(N=100\), of which \(K=30\) are carp. A sample of size \(n=20\) is chosen, and we want to find the mean and variance of the number of carp among these 20 fish caught.

Step 2: Calculate the mean

Mean of a hypergeometric distribution is given by: Mean (\(E[X]\)) = \(n \times \frac{K}{N}\) Where, \(n\) = number of fish caught \(K\) = number of carp in the pond \(N\) = total number of fish in the pond Mean (\(E[X]\)) = \(20 \times \frac{30}{100}\) = \(6\) So, the mean number of carps in the 20 fish caught is 6.

Step 3: Calculate the variance

Variance of a hypergeometric distribution is given by: Variance (\(Var[X]\)) = \(n \times \frac{K}{N} \times \frac{(N-K)}{N} \times \frac{(N-n)}{(N-1)}\) Where, \(n\) = number of fish caught \(K\) = number of carp in the pond \(N\) = total number of fish in the pond Variance (\(Var[X]\)) = \(20 \times \frac{30}{100} \times \frac{(100-30)}{100} \times \frac{(100-20)}{(100-1)}\) = \(20 \times \frac{30}{100} \times \frac{70}{100} \times \frac{80}{99}\) Variance (\(Var[X]\)) = \(2.8282\) So, the variance of the number of carps in the 20 fish caught is approximately 2.8282. In conclusion, when catching 20 fish from the pond, the mean number of carps among them is 6, and the variance is approximately 2.8282, under the assumption that each fish has an equal chance of being caught and we catch each fish independently.

Key Concepts

Understanding Probability Theory

Within the realms of mathematics, probability theory is a monumental pillar that explores the likelihood of events occurring. It serves as the backbone for various studies such as statistics, finance, gambling, science, and even philosophy. When we model real-life scenarios, such as catching fish from a pond, we are employing concepts from probability theory to quantify uncertainty and make informed predictions.

The hypergeometric distribution, used in the example problem, is one specific probability distribution which describes the probabilities associated with the number of successes in a sample drawn without replacement from a finite population. This aligns with our fish-catching scenario; because once a fish is caught, it can't be caught again. Hence, we assume the fish are not being replaced in the pond, making each catch a unique event within our model. The application of probability theory in such examples enables us to analyze and interpret real-world situations through a mathematical lens.

Diving into Mean and Variance

The mean and variance are critical statistical measures that provide insights into a distribution's central tendency and variability, respectively. In the context of the hypergeometric problem at hand, the mean (expected value) gives us the average number of carps we expect to catch in our sample of fish. It is a way of summarizing the possible outcomes into one 'expected' result.

On the other hand, the variance measures how much the number of carps caught is likely to fluctuate from the mean value. A higher variance signifies a wider spread of possible outcomes, while a lower variance indicates they are more clustered around the mean. In the fishing scenario, a variance of 2.8282 tells us that the number of carps caught in different trials of catching 20 fish each may slightly vary, but not extensively so. The calculations of these two measures are inherently tied to the assumptions of our probability model, which assumes no replacements and an equal chance of catching any fish.

Exploring Discrete Probability Distributions

Probability distributions are mathematical functions that describe the likelihood of various outcomes. When dealing with countable outcomes, like the number of carps caught from a pond, we utilize discrete probability distributions. Each potential outcome has a specific probability attached to it.

The hypergeometric distribution, as seen in our exercise, is a type of discrete distribution important for scenarios where we have a finite population and select a sample without replacement. This contrasts with the binomial distribution, which does allow for replacement and requires each trial to be independent. The key feature of any discrete distribution is that it gives us a powerful tool to calculate the probabilities of precise outcomes rather than generalizations. Through these calculations, we are equipped to understand and predict the patterns and behaviors observable in discrete random events.