Answer :
Certainly! Let's summarize the steps and detailed calculations based on your prompt and the recorded data:
### Data Collection
A group of fishermen caught and released 100 fish three times, recording the number of tagged fish each time. The recorded data was:
- Day 1: 20 tagged fish
- Day 2: 25 tagged fish
- Day 3: 15 tagged fish
### Calculations
1. Experimental Probability of Catching a Tagged Fish:
For each day, the experimental probability of catching a tagged fish is calculated by dividing the number of tagged fish caught by the total number of fish caught.
- Day 1:
[tex]\[ \text{Probability} = \frac{\text{Number of tagged fish}}{\text{Total fish caught}} = \frac{20}{100} = 0.2 \][/tex]
- Day 2:
[tex]\[ \text{Probability} = \frac{\text{Number of tagged fish}}{\text{Total fish caught}} = \frac{25}{100} = 0.25 \][/tex]
- Day 3:
[tex]\[ \text{Probability} = \frac{\text{Number of tagged fish}}{\text{Total fish caught}} = \frac{15}{100} = 0.15 \][/tex]
2. Estimated Number of Fish in the Pond:
Assume the total number of tagged fish initially released into the pond is [tex]\( t = 50 \)[/tex]. The estimated population [tex]\( N \)[/tex] for each day can be calculated using the formula:
[tex]\[ N = \frac{\text{Total fish caught} \times t}{\text{Number of tagged fish caught}} \][/tex]
- Day 1:
[tex]\[ N = \frac{100 \times 50}{20} = \frac{5000}{20} = 250 \][/tex]
- Day 2:
[tex]\[ N = \frac{100 \times 50}{25} = \frac{5000}{25} = 200 \][/tex]
- Day 3:
[tex]\[ N = \frac{100 \times 50}{15} = \frac{5000}{15} \approx 333.33 \approx 333 \ \text{(Rounding to the nearest fish)} \][/tex]
### Summary of Results
- Day 1
- Number of tagged fish: 20
- Experimental Probability: 0.2
- Estimated Population: 250
- Day 2
- Number of tagged fish: 25
- Experimental Probability: 0.25
- Estimated Population: 200
- Day 3
- Number of tagged fish: 15
- Experimental Probability: 0.15
- Estimated Population: 333
### Consistency of Estimated Populations
The estimated populations for the three days are:
- Day 1: 250 fish
- Day 2: 200 fish
- Day 3: 333 fish
These estimates vary and do not show a high level of consistency. The variations can occur due to randomness in sampling and the inherent variation expected in probabilistic events. Hence, another method or a larger sample size might be needed to provide more consistent estimates.
### Data Collection
A group of fishermen caught and released 100 fish three times, recording the number of tagged fish each time. The recorded data was:
- Day 1: 20 tagged fish
- Day 2: 25 tagged fish
- Day 3: 15 tagged fish
### Calculations
1. Experimental Probability of Catching a Tagged Fish:
For each day, the experimental probability of catching a tagged fish is calculated by dividing the number of tagged fish caught by the total number of fish caught.
- Day 1:
[tex]\[ \text{Probability} = \frac{\text{Number of tagged fish}}{\text{Total fish caught}} = \frac{20}{100} = 0.2 \][/tex]
- Day 2:
[tex]\[ \text{Probability} = \frac{\text{Number of tagged fish}}{\text{Total fish caught}} = \frac{25}{100} = 0.25 \][/tex]
- Day 3:
[tex]\[ \text{Probability} = \frac{\text{Number of tagged fish}}{\text{Total fish caught}} = \frac{15}{100} = 0.15 \][/tex]
2. Estimated Number of Fish in the Pond:
Assume the total number of tagged fish initially released into the pond is [tex]\( t = 50 \)[/tex]. The estimated population [tex]\( N \)[/tex] for each day can be calculated using the formula:
[tex]\[ N = \frac{\text{Total fish caught} \times t}{\text{Number of tagged fish caught}} \][/tex]
- Day 1:
[tex]\[ N = \frac{100 \times 50}{20} = \frac{5000}{20} = 250 \][/tex]
- Day 2:
[tex]\[ N = \frac{100 \times 50}{25} = \frac{5000}{25} = 200 \][/tex]
- Day 3:
[tex]\[ N = \frac{100 \times 50}{15} = \frac{5000}{15} \approx 333.33 \approx 333 \ \text{(Rounding to the nearest fish)} \][/tex]
### Summary of Results
- Day 1
- Number of tagged fish: 20
- Experimental Probability: 0.2
- Estimated Population: 250
- Day 2
- Number of tagged fish: 25
- Experimental Probability: 0.25
- Estimated Population: 200
- Day 3
- Number of tagged fish: 15
- Experimental Probability: 0.15
- Estimated Population: 333
### Consistency of Estimated Populations
The estimated populations for the three days are:
- Day 1: 250 fish
- Day 2: 200 fish
- Day 3: 333 fish
These estimates vary and do not show a high level of consistency. The variations can occur due to randomness in sampling and the inherent variation expected in probabilistic events. Hence, another method or a larger sample size might be needed to provide more consistent estimates.
