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Sagot :
Certainly! Let’s find the probability that the block is a cube given that it is green. We need to use the concept of conditional probability for this. Here is a step-by-step solution:
### Step-by-Step Solution
1. Define the Given Probabilities:
- The probability that the block is a cube, [tex]\( P(\text{Cube}) \)[/tex], is 0.3.
- The probability that the block is green, [tex]\( P(\text{Green}) \)[/tex], is 0.4.
- The probability that the block is a green cube, [tex]\( P(\text{Green} \cap \text{Cube}) \)[/tex], is 0.07.
2. Identify the Conditional Probability Formula:
To find the probability that the block is a cube given that it is green, we use the conditional probability formula:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{P(\text{Green} \cap \text{Cube})}{P(\text{Green})} \][/tex]
3. Substitute the Known Values into the Formula:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{P(\text{Green} \cap \text{Cube})}{P(\text{Green})} = \frac{0.07}{0.4} \][/tex]
4. Divide the Probabilities:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{0.07}{0.4} = 0.175 \][/tex]
5. Express the Result as a Fraction:
We convert the decimal result, 0.175, into a fraction. The decimal 0.175 can be written as a fraction over 1:
[tex]\[ 0.175 = \frac{175}{1000} \][/tex]
6. Simplify the Fraction:
Simplify [tex]\(\frac{175}{1000}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 25:
[tex]\[ \frac{175}{1000} = \frac{175 \div 25}{1000 \div 25} = \frac{7}{40} \][/tex]
### Final Answer
The probability that the block is a cube given that it is green is:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{7}{40} \][/tex]
Thus, given that the block is green, the probability that it is a cube is [tex]\(\frac{7}{40}\)[/tex] in its simplest form.
### Step-by-Step Solution
1. Define the Given Probabilities:
- The probability that the block is a cube, [tex]\( P(\text{Cube}) \)[/tex], is 0.3.
- The probability that the block is green, [tex]\( P(\text{Green}) \)[/tex], is 0.4.
- The probability that the block is a green cube, [tex]\( P(\text{Green} \cap \text{Cube}) \)[/tex], is 0.07.
2. Identify the Conditional Probability Formula:
To find the probability that the block is a cube given that it is green, we use the conditional probability formula:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{P(\text{Green} \cap \text{Cube})}{P(\text{Green})} \][/tex]
3. Substitute the Known Values into the Formula:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{P(\text{Green} \cap \text{Cube})}{P(\text{Green})} = \frac{0.07}{0.4} \][/tex]
4. Divide the Probabilities:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{0.07}{0.4} = 0.175 \][/tex]
5. Express the Result as a Fraction:
We convert the decimal result, 0.175, into a fraction. The decimal 0.175 can be written as a fraction over 1:
[tex]\[ 0.175 = \frac{175}{1000} \][/tex]
6. Simplify the Fraction:
Simplify [tex]\(\frac{175}{1000}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 25:
[tex]\[ \frac{175}{1000} = \frac{175 \div 25}{1000 \div 25} = \frac{7}{40} \][/tex]
### Final Answer
The probability that the block is a cube given that it is green is:
[tex]\[ P(\text{Cube} \mid \text{Green}) = \frac{7}{40} \][/tex]
Thus, given that the block is green, the probability that it is a cube is [tex]\(\frac{7}{40}\)[/tex] in its simplest form.
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