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Sagot :
To solve this problem, let's go through each statement and determine if it accurately represents the situation based on the given data.
First, let's repeat the data provided in the table:
- 2 is rolled 16 times.
- 3 is rolled 14 times.
- 4 is rolled 20 times.
- 5 is rolled 12 times.
- 6 is rolled 17 times.
The total number of outcomes is:
[tex]\[ 16 + 14 + 20 + 12 + 17 = 79 \][/tex]
Next, let's evaluate each of the statements.
1. The relative frequency of rolling a 4 is [tex]\(\frac{2}{9}\)[/tex].
- The relative frequency of rolling a 4 is calculated as the number of times 4 is rolled divided by the total number of outcomes:
[tex]\[ \text{Relative frequency of 4} = \frac{20}{79} \][/tex]
- We need to check if [tex]\(\frac{20}{79}\)[/tex] is equal to [tex]\(\frac{2}{9}\)[/tex]. Since
[tex]\(\frac{20}{79}\)[/tex] is not equal to [tex]\(\frac{2}{9}\)[/tex], this statement is False.
2. The experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3.
- The experimental probability of rolling a 3 is:
[tex]\[ \text{Experimental probability of 3} = \frac{14}{79} \][/tex]
- The theoretical probability of rolling a 3 with a fair 6-sided die is:
[tex]\[ \text{Theoretical probability of 3} = \frac{1}{6} \][/tex]
- Comparing [tex]\(\frac{14}{79}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex], we find that [tex]\(\frac{14}{79}\)[/tex] is indeed greater than [tex]\(\frac{1}{6}\)[/tex]. Therefore, this statement is True.
3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.
- The experimental probability of rolling a 2 is:
[tex]\[ \text{Experimental probability of 2} = \frac{16}{79} \][/tex]
- The theoretical probability of rolling a 2 is:
[tex]\[ \text{Theoretical probability of 2} = \frac{1}{6} \][/tex]
- Since [tex]\(\frac{16}{79}\)[/tex] is greater than [tex]\(\frac{1}{6}\)[/tex], this statement is True.
4. The relative frequency of rolling a 5 is [tex]\(\frac{2}{13}\)[/tex].
- The relative frequency of rolling a 5 is:
[tex]\[ \text{Relative frequency of 5} = \frac{12}{79} \][/tex]
- We need to check if [tex]\(\frac{12}{79}\)[/tex] is equal to [tex]\(\frac{2}{13}\)[/tex]. Since
[tex]\(\frac{12}{79}\)[/tex] is not equal to [tex]\(\frac{2}{13}\)[/tex], this statement is False.
5. The experimental probability of rolling a 1 is less than the experimental probability of rolling a 6.
- From the given data, a 1 is never rolled, so:
[tex]\[ \text{Experimental probability of 1} = 0 \][/tex]
- The experimental probability of rolling a 6 is:
[tex]\[ \text{Experimental probability of 6} = \frac{17}{79} \][/tex]
- Since [tex]\(0\)[/tex] is certainly less than [tex]\(\frac{17}{79}\)[/tex], this statement is True.
Given the evaluations, the three correct statements representing the situation are:
- The experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3. (True)
- The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2. (True)
- The experimental probability of rolling a 1 is less than the experimental probability of rolling a 6. (True)
First, let's repeat the data provided in the table:
- 2 is rolled 16 times.
- 3 is rolled 14 times.
- 4 is rolled 20 times.
- 5 is rolled 12 times.
- 6 is rolled 17 times.
The total number of outcomes is:
[tex]\[ 16 + 14 + 20 + 12 + 17 = 79 \][/tex]
Next, let's evaluate each of the statements.
1. The relative frequency of rolling a 4 is [tex]\(\frac{2}{9}\)[/tex].
- The relative frequency of rolling a 4 is calculated as the number of times 4 is rolled divided by the total number of outcomes:
[tex]\[ \text{Relative frequency of 4} = \frac{20}{79} \][/tex]
- We need to check if [tex]\(\frac{20}{79}\)[/tex] is equal to [tex]\(\frac{2}{9}\)[/tex]. Since
[tex]\(\frac{20}{79}\)[/tex] is not equal to [tex]\(\frac{2}{9}\)[/tex], this statement is False.
2. The experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3.
- The experimental probability of rolling a 3 is:
[tex]\[ \text{Experimental probability of 3} = \frac{14}{79} \][/tex]
- The theoretical probability of rolling a 3 with a fair 6-sided die is:
[tex]\[ \text{Theoretical probability of 3} = \frac{1}{6} \][/tex]
- Comparing [tex]\(\frac{14}{79}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex], we find that [tex]\(\frac{14}{79}\)[/tex] is indeed greater than [tex]\(\frac{1}{6}\)[/tex]. Therefore, this statement is True.
3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.
- The experimental probability of rolling a 2 is:
[tex]\[ \text{Experimental probability of 2} = \frac{16}{79} \][/tex]
- The theoretical probability of rolling a 2 is:
[tex]\[ \text{Theoretical probability of 2} = \frac{1}{6} \][/tex]
- Since [tex]\(\frac{16}{79}\)[/tex] is greater than [tex]\(\frac{1}{6}\)[/tex], this statement is True.
4. The relative frequency of rolling a 5 is [tex]\(\frac{2}{13}\)[/tex].
- The relative frequency of rolling a 5 is:
[tex]\[ \text{Relative frequency of 5} = \frac{12}{79} \][/tex]
- We need to check if [tex]\(\frac{12}{79}\)[/tex] is equal to [tex]\(\frac{2}{13}\)[/tex]. Since
[tex]\(\frac{12}{79}\)[/tex] is not equal to [tex]\(\frac{2}{13}\)[/tex], this statement is False.
5. The experimental probability of rolling a 1 is less than the experimental probability of rolling a 6.
- From the given data, a 1 is never rolled, so:
[tex]\[ \text{Experimental probability of 1} = 0 \][/tex]
- The experimental probability of rolling a 6 is:
[tex]\[ \text{Experimental probability of 6} = \frac{17}{79} \][/tex]
- Since [tex]\(0\)[/tex] is certainly less than [tex]\(\frac{17}{79}\)[/tex], this statement is True.
Given the evaluations, the three correct statements representing the situation are:
- The experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3. (True)
- The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2. (True)
- The experimental probability of rolling a 1 is less than the experimental probability of rolling a 6. (True)
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