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Sagot :
To analyze the scenario of selecting a marble from the bag, let's first summarize the contents of the bag:
- Number of red marbles: 4
- Number of yellow marbles: 3
- Number of purple marbles: 3
Total number of marbles in the bag: [tex]\( 4 + 3 + 3 = 10 \)[/tex]
We will now determine the probability of selecting each type of marble.
1. The probability you select a red marble is [tex]\(\frac{4}{10}\)[/tex]:
The probability can be computed by taking the number of red marbles and dividing it by the total number of marbles:
[tex]\[ P(\text{Red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{4}{10} = 0.4 \][/tex]
Therefore, this statement is false because it states the probability as [tex]\(\frac{3}{10}\)[/tex], but the correct probability is [tex]\(\frac{4}{10}\)[/tex], which is 0.4.
2. The probability you select a purple marble is [tex]\(\frac{3}{10}\)[/tex]:
We compute it similarly by dividing the number of purple marbles by the total number of marbles:
[tex]\[ P(\text{Purple}) = \frac{\text{Number of purple marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]
Therefore, this statement is true.
3. [tex]\( P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1 \)[/tex]:
The probabilities of red, yellow, and purple marbles should sum up to 1 as they are the only options available in the bag.
[tex]\[ P(\text{Red}) = 0.4, \quad P(\text{Yellow}) = \frac{3}{10} = 0.3, \quad P(\text{Purple}) = 0.3 \][/tex]
Adding them together:
[tex]\[ P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 0.4 + 0.3 + 0.3 = 1.0 \][/tex]
Therefore, this statement is true.
4. The probability you select a green marble is 0:
Since there are no green marbles in the bag:
[tex]\[ P(\text{Green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{0}{10} = 0 \][/tex]
Therefore, this statement is true.
5. The probability you select a yellow marble is 3:
The correct probability of selecting a yellow marble is:
[tex]\[ P(\text{Yellow}) = \frac{\text{Number of yellow marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]
Therefore, this statement is false because it incorrectly gives the probability as 3, while the correct value is 0.3.
So, the true statements are:
- The probability you select a purple marble is [tex]\(\frac{3}{10}\)[/tex].
- [tex]\( P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1 \)[/tex]
- The probability you select a green marble is 0.
- Number of red marbles: 4
- Number of yellow marbles: 3
- Number of purple marbles: 3
Total number of marbles in the bag: [tex]\( 4 + 3 + 3 = 10 \)[/tex]
We will now determine the probability of selecting each type of marble.
1. The probability you select a red marble is [tex]\(\frac{4}{10}\)[/tex]:
The probability can be computed by taking the number of red marbles and dividing it by the total number of marbles:
[tex]\[ P(\text{Red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{4}{10} = 0.4 \][/tex]
Therefore, this statement is false because it states the probability as [tex]\(\frac{3}{10}\)[/tex], but the correct probability is [tex]\(\frac{4}{10}\)[/tex], which is 0.4.
2. The probability you select a purple marble is [tex]\(\frac{3}{10}\)[/tex]:
We compute it similarly by dividing the number of purple marbles by the total number of marbles:
[tex]\[ P(\text{Purple}) = \frac{\text{Number of purple marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]
Therefore, this statement is true.
3. [tex]\( P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1 \)[/tex]:
The probabilities of red, yellow, and purple marbles should sum up to 1 as they are the only options available in the bag.
[tex]\[ P(\text{Red}) = 0.4, \quad P(\text{Yellow}) = \frac{3}{10} = 0.3, \quad P(\text{Purple}) = 0.3 \][/tex]
Adding them together:
[tex]\[ P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 0.4 + 0.3 + 0.3 = 1.0 \][/tex]
Therefore, this statement is true.
4. The probability you select a green marble is 0:
Since there are no green marbles in the bag:
[tex]\[ P(\text{Green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{0}{10} = 0 \][/tex]
Therefore, this statement is true.
5. The probability you select a yellow marble is 3:
The correct probability of selecting a yellow marble is:
[tex]\[ P(\text{Yellow}) = \frac{\text{Number of yellow marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]
Therefore, this statement is false because it incorrectly gives the probability as 3, while the correct value is 0.3.
So, the true statements are:
- The probability you select a purple marble is [tex]\(\frac{3}{10}\)[/tex].
- [tex]\( P(\text{Red}) + P(\text{Yellow}) + P(\text{Purple}) = 1 \)[/tex]
- The probability you select a green marble is 0.
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