Problem 33 - A First Course in Probability

Question

A forest contains 20 elk, of which 5 are captured, tagged, and then released. A certain time later, 4 of the 20 elk are captured. What is the probability that 2 of these 4 have been tagged? What assumptions are you making?

Step-by-Step Solution

Answer

The probability that 2 out of the 4 captured elk have been tagged is \(\frac{210}{969}\), assuming that all elk have an equal chance of being captured and the tagging event does not affect their likelihood of being captured again.

Step 1: Understand the problem and identify given data

We are given: - A forest containing 20 elk. - 5 elk have been tagged and then released. - A certain time later, 4 elk are captured. - We need to find the probability that 2 of these 4 have been tagged.

Step 2: Calculate the total combinations

We will first find the total number of ways of selecting 4 elk out of 20 elk. This can be found using the formula for combinations: \(C(n, k) = \frac{n!}{k!(n-k)!}\) Here, n = 20 (total elk) and k = 4 (captured elk). Total Combinations: \(C(20, 4) = \frac{20!}{4!16!} = 4845\)

Step 3: Calculate the combinations of 2 tagged elk and 2 non-tagged elk

Now, we will find the number of combinations in which 2 out of the 4 elk are tagged (from the 5 tagged elk) and 2 are not tagged (from the remaining 15 elk). Tagged Elk Combinations: \(C(5, 2) = \frac{5!}{2!3!} = 10\) Non-Tagged Elk Combinations: \(C(15, 2) = \frac{15!}{2!13!} = 105\) Now we will find the product of tagged elk combinations and non-tagged elk combinations: Product of Combinations: \(10 \times 105 = 1050\)

Step 4: Determine the probability

We will now determine the probability by dividing the product of combinations by the total combinations: Probability: \(\frac{Product \thinspace of \thinspace Combinations}{Total \thinspace Combinations} = \frac{1050}{4845} = \frac{210}{969}\)

Step 5: Assumptions made

Throughout the solution, the following assumptions have been made: 1. All elk have an equal chance of being captured. 2. The first capture and tagging event does not affect the behavior or likelihood of an elk being captured again later. Hence, the probability that 2 out of the 4 captured elk have been tagged is \(\frac{210}{969}\).

Key Concepts

Combinations

In probability, combinations are used when we need to determine how many different ways we can select a specific number of items from a larger group, without regard to the order of the items. This is very useful when analyzing problems where the sequence does not matter, like drawing cards from a deck or selecting elk from a forest.
The formula for combinations is given by:

• \[ C(n, k) = \frac{n!}{k!(n-k)!} \]

Here, \(n\) is the total number of items, and \(k\) is the number of items to choose.
For instance, with 20 elk in a forest and 4 to be captured, we calculate all possible combinations of selecting 4 elk: \[ C(20, 4) = \frac{20!}{4!16!} = 4845 \] ways. This calculation tells us there are 4845 unique groups of 4 elk that can be chosen from the total of 20. Each unique group or selection is called a combination.

Tagged Elk

Tagged Elk refer to the elk that have been marked or tagged for research or tracking purposes. In our scenario, 5 out of 20 elk are tagged and released back into the wild.
Understanding the role of tagged elk is essential because it helps us determine how likely these specific individuals are to be part of future captures.
In our probability problem, we seek to find out the likelihood that exactly 2 out of the 4 captured elk are part of this special group of tagged elk.
To find combinations involving tagged elk, we use:

• \[ C(5, 2) = \frac{5!}{2!3!} = 10 \]

This formula calculates the number of ways to choose 2 elk from the 5 that are tagged.

Equal Probability

The assumption of equal probability is vital for simplifying our calculations in probability problems. It means each elk has the same chance of being captured during sampling, regardless of its previous status or interactions.
In our exercise, we assume each elk is equally likely to be one of the 4 captured, which affects our calculation of probabilities.
This ensures that the tagging does not influence the elk's chance of being taken again.
This kind of equal probability is crucial for methods like combinatorial probability because it allows us to evenly consider all possibilities when forming combinations.

Sampling Without Replacement

Sampling without replacement is a method where each item, once selected, is not returned to the general pool to be potentially selected again.
This differs from sampling with replacement, where items can be picked multiple times.
In the context of our elk problem, once we capture an elk, it is not available for selection again in this round of sampling.
This affects the calculation of probability. The total number of elk reduces with each capture, making the available pool smaller with every selection.
Thus, the probabilities change dynamically, reflecting the removal of items from the population, leading to calculations like the probability of capturing 2 tagged elk as \(\frac{210}{969}\).