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Chapter 10: Problem 2

Last month it was estimated that a lake contained 3500 rainbow trout. Over a three-day period a park ranger caught, tagged, and released 100 fish. Then, after allowing two weeks for random mixing, she caught 100 more rainbow trout and found that 3 of them had tags. a. What is the probability of catching a tagged trout? b. What assumptions must you make to answer \(2 \mathrm{a}\) ? c. Based on the number of tagged fish she caught two weeks later, what is the park ranger's experimental probability?

Short Answer

Expert verified
The probability is 0.03, assuming random mixing and stable population.

Step by step solution

01

Understanding the Problem

We begin with a population of 3500 rainbow trout and initially tag 100 of them. After two weeks, 100 trout are caught again, and 3 are found to have tags.
02

Calculate Probability of Catching a Tagged Trout

The probability of catching a tagged trout is calculated by dividing the number of tagged trout caught by the total number of trout caught: number of tagged trout caught is 3, and the total caught again is 100. Thus, the probability is \( \frac{3}{100} = 0.03 \).
03

State Assumptions

We assume that the tagged trout have mixed fully with the population and that each trout has an equal probability of being caught again. We also assume no change in the population size of the trout lake during this time (no births, deaths, or migrations).
04

Calculate Experimental Probability

To calculate the experimental probability, divide the number of tagged fish caught by the total caught two weeks later: 3 tagged fish out of 100 total fish gives an experimental probability of \( \frac{3}{100} = 0.03 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tagging Method
The tagging method is a common technique used by researchers to study the size and behavior of wildlife populations. It involves capturing a sample of individuals from the population, attaching a unique identifier or tag to each, and then releasing them back into their environment.
This method allows scientists to gather data on the movement patterns, survival rates, and population size by later recapturing some of the tagged individuals.
In the context of our exercise, the park ranger uses tagging to help estimate the population of rainbow trout.
  • Initially, 100 trout are caught, tagged, and released.
  • This step is crucial as it marks a known number of individuals within the larger group.
By tagging and then recapturing a sample later, the ranger can use probability to estimate the number of fish in the entire lake.
Population Estimation
Population estimation using the tagging method is based on the principle of proportionality. This means that the ratio of tagged to untagged fish in a sample should reflect the ratio of tagged to untagged fish in the entire population.
In our example, the ranger tagged 100 fish out of an estimated 3500. When she later caught 100 trout, and 3 were tagged, these numbers help estimate the total population.
  • If the ratio of tagged fish in the sample (3 out of 100) is the same as in the lake, we can set up the proportion \( \frac{3}{100} = \frac{100}{3500} \).
  • This provides an estimate confirming the initial population size assumption of approximately 3500 trout.
It's a practical way to estimate population size without needing to count every individual directly.
Probability Assumptions
Several assumptions must be made when using tagging for population estimation and probability calculation.
First and foremost, it's assumed that the tagged and untagged fish mix thoroughly across the population. This ensures that each fish, regardless of being tagged or not, has an equal chance of being caught when a sample is taken.
Additionally, assumptions include:
  • No significant change in population size due to deaths, births, or migrations during the study period.
  • Completely random sampling during both tagging and recapture stages.
If these assumptions hold true, the findings (such as a 3% probability of catching a tagged fish) become reliable and can be used for further analysis.
Random Sampling
Random sampling is at the core of accurately estimating populations or probabilities in tagging studies. It ensures that each individual in the population has an equal chance of being selected during the sampling process.
This concept is crucial for generating data that represent the entire community rather than a biased subset.
  • In the ranger's study, both the initial capture for tagging and the recapture sampling process should be random.
  • A truly random sample allows the results, like the 3 out of 100 tagged trout captured later, to indicate the distribution of tagged fish across the entire population accurately.
Without random sampling, the study could lead to skewed data, which might not reflect reality, potentially leading to incorrect conclusions about the population size or tagging success.

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Most popular questions from this chapter

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