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Sagot :
To write a probability model for choosing a bead from the crate, we need to determine the probability of selecting each type of bead. Here is a detailed step-by-step solution:
### Step 1: Determine the total number of beads.
First, we need to find the total number of beads by adding the counts of each type of bead:
- Copper beads: 60
- Wood beads: 75
- Silver beads: 240
The total number of beads is:
[tex]\[ \text{Total beads} = 60 + 75 + 240 = 375 \][/tex]
### Step 2: Calculate the probability of choosing each type of bead.
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the probability of choosing each type of bead is calculated as:
#### Probability of selecting a copper bead:
[tex]\[ P(\text{Copper}) = \frac{\text{Number of Copper beads}}{\text{Total number of beads}} = \frac{60}{375} \][/tex]
#### Probability of selecting a wood bead:
[tex]\[ P(\text{Wood}) = \frac{\text{Number of Wood beads}}{\text{Total number of beads}} = \frac{75}{375} \][/tex]
#### Probability of selecting a silver bead:
[tex]\[ P(\text{Silver}) = \frac{\text{Number of Silver beads}}{\text{Total number of beads}} = \frac{240}{375} \][/tex]
### Step 3: Simplify the fractions.
We can simplify these fractions by finding the greatest common divisor (GCD) for each fraction:
#### Simplifying [tex]\( \frac{60}{375} \)[/tex]:
The GCD of 60 and 375 is 15:
[tex]\[ \frac{60 \div 15}{375 \div 15} = \frac{4}{25} \][/tex]
#### Simplifying [tex]\( \frac{75}{375} \)[/tex]:
The GCD of 75 and 375 is 75:
[tex]\[ \frac{75 \div 75}{375 \div 75} = \frac{1}{5} \][/tex]
#### Simplifying [tex]\( \frac{240}{375} \)[/tex]:
The GCD of 240 and 375 is 15:
[tex]\[ \frac{240 \div 15}{375 \div 15} = \frac{16}{25} \][/tex]
### Step 4: Verify the probabilities sum to 1.
The sum of all probabilities should equal 1 to ensure the model is valid:
[tex]\[ P(\text{Copper}) + P(\text{Wood}) + P(\text{Silver}) = \frac{4}{25} + \frac{5}{25} + \frac{16}{25} = \frac{25}{25} = 1 \][/tex]
### Step 5: Present the probability model.
Here is the probability model for choosing a bead from the crate:
- Probability of choosing a copper bead: [tex]\( \frac{4}{25} \)[/tex]
- Probability of choosing a wood bead: [tex]\( \frac{1}{5} \)[/tex]
- Probability of choosing a silver bead: [tex]\( \frac{16}{25} \)[/tex]
These probabilities sum up to 1, confirming that the probability model is correct and valid.
### Step 1: Determine the total number of beads.
First, we need to find the total number of beads by adding the counts of each type of bead:
- Copper beads: 60
- Wood beads: 75
- Silver beads: 240
The total number of beads is:
[tex]\[ \text{Total beads} = 60 + 75 + 240 = 375 \][/tex]
### Step 2: Calculate the probability of choosing each type of bead.
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the probability of choosing each type of bead is calculated as:
#### Probability of selecting a copper bead:
[tex]\[ P(\text{Copper}) = \frac{\text{Number of Copper beads}}{\text{Total number of beads}} = \frac{60}{375} \][/tex]
#### Probability of selecting a wood bead:
[tex]\[ P(\text{Wood}) = \frac{\text{Number of Wood beads}}{\text{Total number of beads}} = \frac{75}{375} \][/tex]
#### Probability of selecting a silver bead:
[tex]\[ P(\text{Silver}) = \frac{\text{Number of Silver beads}}{\text{Total number of beads}} = \frac{240}{375} \][/tex]
### Step 3: Simplify the fractions.
We can simplify these fractions by finding the greatest common divisor (GCD) for each fraction:
#### Simplifying [tex]\( \frac{60}{375} \)[/tex]:
The GCD of 60 and 375 is 15:
[tex]\[ \frac{60 \div 15}{375 \div 15} = \frac{4}{25} \][/tex]
#### Simplifying [tex]\( \frac{75}{375} \)[/tex]:
The GCD of 75 and 375 is 75:
[tex]\[ \frac{75 \div 75}{375 \div 75} = \frac{1}{5} \][/tex]
#### Simplifying [tex]\( \frac{240}{375} \)[/tex]:
The GCD of 240 and 375 is 15:
[tex]\[ \frac{240 \div 15}{375 \div 15} = \frac{16}{25} \][/tex]
### Step 4: Verify the probabilities sum to 1.
The sum of all probabilities should equal 1 to ensure the model is valid:
[tex]\[ P(\text{Copper}) + P(\text{Wood}) + P(\text{Silver}) = \frac{4}{25} + \frac{5}{25} + \frac{16}{25} = \frac{25}{25} = 1 \][/tex]
### Step 5: Present the probability model.
Here is the probability model for choosing a bead from the crate:
- Probability of choosing a copper bead: [tex]\( \frac{4}{25} \)[/tex]
- Probability of choosing a wood bead: [tex]\( \frac{1}{5} \)[/tex]
- Probability of choosing a silver bead: [tex]\( \frac{16}{25} \)[/tex]
These probabilities sum up to 1, confirming that the probability model is correct and valid.
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