To estimate the total snail population in the woodland, we can use a method known as the Lincoln-Petersen Index. This method relies on the idea of marking a portion of the population, releasing them, and then recapturing a sample to see how many of the marked individuals are recaptured. Here's the step-by-step solution:
1. Initial Marking:
- The biologist initially captured and marked 84 snails. This means there are 84 marked snails in the population.
2. Second Capture:
- Fifteen days later, the biologist captured another sample of 90 snails from the woodland.
- Out of these 90 snails captured the second time, 12 were found to be marked.
3. Setting Up a Proportion:
- We can set up a proportion to estimate the total population \( N \) based on the idea that the ratio of marked snails to the total population should be approximately equal to the ratio of marked snails in the second capture to the total number of snails captured in the second sample.
- In mathematical terms, this can be written as:
[tex]\[
\frac{\text{marked snails initially}}{\text{total population}} \approx \frac{\text{marked snails in second capture}}{\text{total snails in second capture}}
\][/tex]
4. Plugging in the Values:
- Let's denote the total snail population as \( N \).
- According to the problem:
[tex]\[
\frac{84}{N} = \frac{12}{90}
\][/tex]
5. Solving for \( N \):
- Cross multiply to solve for \( N \):
[tex]\[
84 \times 90 = 12 \times N
\][/tex]
[tex]\[
7560 = 12N
\][/tex]
[tex]\[
N = \frac{7560}{12}
\][/tex]
[tex]\[
N = 630
\][/tex]
Therefore, based on the probabilities suggested by this experiment, the biologist should estimate that the total snail population in the woodland is 630.
