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Sagot :
Let's analyze Mary's claim step-by-step and determine why she is incorrect.
1. Understanding Experimental Probability:
The experimental probability [tex]\( P \)[/tex] of an event is calculated using the formula:
[tex]\[ P = \frac{\text{Number of successful outcomes}}{\text{Total number of trials}} \][/tex]
2. Given Data:
- Successful outcomes (landing on green): 9 times
- Total number of trials: 45 times
We will use these values to calculate the experimental probability.
3. Calculation of Experimental Probability:
[tex]\[ P = \frac{9}{45} \][/tex]
Simplify the fraction [tex]\(\frac{9}{45}\)[/tex]:
- The Greatest Common Divisor (GCD) of 9 and 45 is 9.
[tex]\[ \frac{9}{45} = \frac{9 \div 9}{45 \div 9} = \frac{1}{5} \][/tex]
4. Interpreting the Result:
The simplified experimental probability is [tex]\(\frac{1}{5}\)[/tex]. This means that in this experiment, the probability of the spinner landing on green is [tex]\(\frac{1}{5}\)[/tex].
5. Analyzing Mary’s Statement:
- Mary claims the experimental probability was [tex]\(\frac{1}{6}\)[/tex].
- Based on our calculation, we established that the actual experimental probability is [tex]\(\frac{1}{5}\)[/tex], not [tex]\(\frac{1}{6}\)[/tex].
6. Why Mary is Incorrect:
Mary is incorrect because:
- The actual simplified experimental probability is [tex]\(\frac{1}{5}\)[/tex], which does not match her claim of [tex]\(\frac{1}{6}\)[/tex].
- If Mary calculated the experimental probability as [tex]\(\frac{1}{6}\)[/tex], it suggests a misunderstanding in either the simplification process or the initial values used.
7. Possible Reasons for Mary’s Miscalculation:
- Mary might have made a multiplication error, possibly multiplying the 9 occurrences by an incorrect factor (e.g., thinking 9 occurs every 6 times instead of every 5).
- Mary could have mistakenly thought that [tex]\(\frac{1}{6}\)[/tex] was the simplest form of a fraction that doesn't actually simplify to [tex]\(\frac{1}{6}\)[/tex] (e.g., [tex]\(\frac{6}{36}\)[/tex] or similar less apparent errors).
To sum up, Mary’s error lies in the miscalculation of the experimental probability. With 9 occurrences out of 45 trials, the correct simplified experimental probability is [tex]\(\frac{1}{5}\)[/tex], not [tex]\(\frac{1}{6}\)[/tex]. This discrepancy invalidates Mary's statement and illustrates her incorrect reasoning.
1. Understanding Experimental Probability:
The experimental probability [tex]\( P \)[/tex] of an event is calculated using the formula:
[tex]\[ P = \frac{\text{Number of successful outcomes}}{\text{Total number of trials}} \][/tex]
2. Given Data:
- Successful outcomes (landing on green): 9 times
- Total number of trials: 45 times
We will use these values to calculate the experimental probability.
3. Calculation of Experimental Probability:
[tex]\[ P = \frac{9}{45} \][/tex]
Simplify the fraction [tex]\(\frac{9}{45}\)[/tex]:
- The Greatest Common Divisor (GCD) of 9 and 45 is 9.
[tex]\[ \frac{9}{45} = \frac{9 \div 9}{45 \div 9} = \frac{1}{5} \][/tex]
4. Interpreting the Result:
The simplified experimental probability is [tex]\(\frac{1}{5}\)[/tex]. This means that in this experiment, the probability of the spinner landing on green is [tex]\(\frac{1}{5}\)[/tex].
5. Analyzing Mary’s Statement:
- Mary claims the experimental probability was [tex]\(\frac{1}{6}\)[/tex].
- Based on our calculation, we established that the actual experimental probability is [tex]\(\frac{1}{5}\)[/tex], not [tex]\(\frac{1}{6}\)[/tex].
6. Why Mary is Incorrect:
Mary is incorrect because:
- The actual simplified experimental probability is [tex]\(\frac{1}{5}\)[/tex], which does not match her claim of [tex]\(\frac{1}{6}\)[/tex].
- If Mary calculated the experimental probability as [tex]\(\frac{1}{6}\)[/tex], it suggests a misunderstanding in either the simplification process or the initial values used.
7. Possible Reasons for Mary’s Miscalculation:
- Mary might have made a multiplication error, possibly multiplying the 9 occurrences by an incorrect factor (e.g., thinking 9 occurs every 6 times instead of every 5).
- Mary could have mistakenly thought that [tex]\(\frac{1}{6}\)[/tex] was the simplest form of a fraction that doesn't actually simplify to [tex]\(\frac{1}{6}\)[/tex] (e.g., [tex]\(\frac{6}{36}\)[/tex] or similar less apparent errors).
To sum up, Mary’s error lies in the miscalculation of the experimental probability. With 9 occurrences out of 45 trials, the correct simplified experimental probability is [tex]\(\frac{1}{5}\)[/tex], not [tex]\(\frac{1}{6}\)[/tex]. This discrepancy invalidates Mary's statement and illustrates her incorrect reasoning.
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