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Sagot :
Sure, let me walk you through the solution step-by-step.
1. Hurricane Counts per Trial:
We start by examining each trial to count how many years had hurricanes. A hurricane is represented by the number '1'. Here are the counts for each trial:
- Trial 1: 1 year with a hurricane (2,5,3,2,5,5,1,4,5,2 → one '1' present)
- Trial 2: 5 years with a hurricane (1,1,5,2,2,1,3,1,1,5 → five '1's present)
- Trial 3: 2 years with a hurricane (4,5,4,5,5,4,3,5,1,1 → two '1's present)
- Trial 4: 2 years with a hurricane (1,5,5,5,1,2,2,3,5,3 → two '1's present)
- Trial 5: 1 year with a hurricane (5,1,5,3,5,3,4,5,3,2 → one '1' present)
- Trial 6: 3 years with a hurricane (1,1,5,5,1,4,2,2,3,4 → three '1's present)
- Trial 7: 4 years with a hurricane (2,1,5,3,1,5,1,2,1,4 → four '1's present)
- Trial 8: 2 years with a hurricane (2,4,3,2,4,4,2,1,3,1 → two '1's present)
- Trial 9: 2 years with a hurricane (3,2,1,4,5,3,5,5,1,2 → two '1's present)
- Trial 10: 1 year with a hurricane (3,4,2,4,3,5,2,3,5,1 → one '1' present)
The resulting counts are `[1, 5, 2, 2, 1, 3, 4, 2, 2, 1]`.
2. Count of Trials with At Least 4 Years with a Hurricane:
From the hurricane counts, we see that exactly 2 trials had at least 4 years with hurricanes:
- Trial 2 (5 years with hurricanes)
- Trial 7 (4 years with hurricanes)
3. Total Number of Trials:
The total number of trials conducted is 10.
4. Experimental Probability:
The experimental probability \( P \) that a hurricane would strike the city in at least 4 of the next 10 years is calculated by the formula:
[tex]\[ P = \left(\frac{\text{Number of trials with at least 4 hurricanes}}{\text{Total number of trials}}\right) \times 100\% \][/tex]
Substituting the values from above:
[tex]\[ P = \left(\frac{2}{10}\right) \times 100\% = 20.0\% \][/tex]
5. Comparison to Actual Result:
According to the given data, there was actually 1 year out of 10 in which a hurricane struck. To compare this to Rob's simulation, it's simpler to express the number of hurricanes per 10 years:
- Actual result: 1 year with a hurricane over 10 years.
- Rob's simulation result: average of 2 years with hurricanes over 10 years (as out of 10 trials with a singular count per trial).
In percentage terms, this means:
- Actual probability: 10%
- Simulated probability: 20.0%
So, Rob's simulation predicted double the probability of at least 4 hurricanes than what actually happened over the subsequent 10-year period, showing a difference of -10%.
The comparison indicates Rob's simulation overestimates the occurrence of hurricanes when compared to the actual data. This suggests the actual chance was less than what Rob's model predicted.
1. Hurricane Counts per Trial:
We start by examining each trial to count how many years had hurricanes. A hurricane is represented by the number '1'. Here are the counts for each trial:
- Trial 1: 1 year with a hurricane (2,5,3,2,5,5,1,4,5,2 → one '1' present)
- Trial 2: 5 years with a hurricane (1,1,5,2,2,1,3,1,1,5 → five '1's present)
- Trial 3: 2 years with a hurricane (4,5,4,5,5,4,3,5,1,1 → two '1's present)
- Trial 4: 2 years with a hurricane (1,5,5,5,1,2,2,3,5,3 → two '1's present)
- Trial 5: 1 year with a hurricane (5,1,5,3,5,3,4,5,3,2 → one '1' present)
- Trial 6: 3 years with a hurricane (1,1,5,5,1,4,2,2,3,4 → three '1's present)
- Trial 7: 4 years with a hurricane (2,1,5,3,1,5,1,2,1,4 → four '1's present)
- Trial 8: 2 years with a hurricane (2,4,3,2,4,4,2,1,3,1 → two '1's present)
- Trial 9: 2 years with a hurricane (3,2,1,4,5,3,5,5,1,2 → two '1's present)
- Trial 10: 1 year with a hurricane (3,4,2,4,3,5,2,3,5,1 → one '1' present)
The resulting counts are `[1, 5, 2, 2, 1, 3, 4, 2, 2, 1]`.
2. Count of Trials with At Least 4 Years with a Hurricane:
From the hurricane counts, we see that exactly 2 trials had at least 4 years with hurricanes:
- Trial 2 (5 years with hurricanes)
- Trial 7 (4 years with hurricanes)
3. Total Number of Trials:
The total number of trials conducted is 10.
4. Experimental Probability:
The experimental probability \( P \) that a hurricane would strike the city in at least 4 of the next 10 years is calculated by the formula:
[tex]\[ P = \left(\frac{\text{Number of trials with at least 4 hurricanes}}{\text{Total number of trials}}\right) \times 100\% \][/tex]
Substituting the values from above:
[tex]\[ P = \left(\frac{2}{10}\right) \times 100\% = 20.0\% \][/tex]
5. Comparison to Actual Result:
According to the given data, there was actually 1 year out of 10 in which a hurricane struck. To compare this to Rob's simulation, it's simpler to express the number of hurricanes per 10 years:
- Actual result: 1 year with a hurricane over 10 years.
- Rob's simulation result: average of 2 years with hurricanes over 10 years (as out of 10 trials with a singular count per trial).
In percentage terms, this means:
- Actual probability: 10%
- Simulated probability: 20.0%
So, Rob's simulation predicted double the probability of at least 4 hurricanes than what actually happened over the subsequent 10-year period, showing a difference of -10%.
The comparison indicates Rob's simulation overestimates the occurrence of hurricanes when compared to the actual data. This suggests the actual chance was less than what Rob's model predicted.
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