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 \(\frac{210}{969}\).