Question
Ronnie has a bag with green, red, yellow, and blue marbles. He conducts a probability experiment. For each event in his experiment, Ronnie randomly selects one marble from the bag, records its color, and then replaces it. The results are shown in the table below.
Color Frequency
Green 30
Red 6
Yellow 12
Blue 24
Based on the experimental probability, if there are 12 marbles in the bag, how many of each marble color are in the bag?
Number of green marbles :
Number of red marbles :
Number of yellow marbles :
Number of blue marbles :
Color Frequency
Green 30
Red 6
Yellow 12
Blue 24
Based on the experimental probability, if there are 12 marbles in the bag, how many of each marble color are in the bag?
Number of green marbles :
Number of red marbles :
Number of yellow marbles :
Number of blue marbles :
Answers
GPT-4o mini
To find out how many of each marble color are in the bag based on the experimental probabilities given the frequencies, let’s start by calculating the total number of trials:
\[
\text{Total Frequency} = 30 + 6 + 12 + 24 = 72
\]
Now, we will calculate the experimental probability for each color:
- Probability of selecting a green marble:
\[
P(\text{Green}) = \frac{30}{72}
\]
- Probability of selecting a red marble:
\[
P(\text{Red}) = \frac{6}{72}
\]
- Probability of selecting a yellow marble:
\[
P(\text{Yellow}) = \frac{12}{72}
\]
- Probability of selecting a blue marble:
\[
P(\text{Blue}) = \frac{24}{72}
\]
Next, we know that the total number of marbles in the bag is 12. Therefore, we can express the number of marbles of each color as follows, using the probabilities calculated above:
Let \( G \) be the number of green marbles, \( R \) be the number of red marbles, \( Y \) be the number of yellow marbles, and \( B \) be the number of blue marbles. Each of these can be calculated based on the proportion of the total 12 marbles.
- Number of green marbles:
\[
G = 12 \cdot P(\text{Green}) = 12 \cdot \frac{30}{72} = 12 \cdot \frac{5}{12} = 5
\]
- Number of red marbles:
\[
R = 12 \cdot P(\text{Red}) = 12 \cdot \frac{6}{72} = 12 \cdot \frac{1}{12} = 1
\]
- Number of yellow marbles:
\[
Y = 12 \cdot P(\text{Yellow}) = 12 \cdot \frac{12}{72} = 12 \cdot \frac{1}{6} = 2
\]
- Number of blue marbles:
\[
B = 12 \cdot P(\text{Blue}) = 12 \cdot \frac{24}{72} = 12 \cdot \frac{1}{3} = 4
\]
Now summarizing the number of each color:
- Number of green marbles: **5**
- Number of red marbles: **1**
- Number of yellow marbles: **2**
- Number of blue marbles: **4**
Thus, the final answer is:
Number of green marbles: **5**
Number of red marbles: **1**
Number of yellow marbles: **2**
Number of blue marbles: **4**
\[
\text{Total Frequency} = 30 + 6 + 12 + 24 = 72
\]
Now, we will calculate the experimental probability for each color:
- Probability of selecting a green marble:
\[
P(\text{Green}) = \frac{30}{72}
\]
- Probability of selecting a red marble:
\[
P(\text{Red}) = \frac{6}{72}
\]
- Probability of selecting a yellow marble:
\[
P(\text{Yellow}) = \frac{12}{72}
\]
- Probability of selecting a blue marble:
\[
P(\text{Blue}) = \frac{24}{72}
\]
Next, we know that the total number of marbles in the bag is 12. Therefore, we can express the number of marbles of each color as follows, using the probabilities calculated above:
Let \( G \) be the number of green marbles, \( R \) be the number of red marbles, \( Y \) be the number of yellow marbles, and \( B \) be the number of blue marbles. Each of these can be calculated based on the proportion of the total 12 marbles.
- Number of green marbles:
\[
G = 12 \cdot P(\text{Green}) = 12 \cdot \frac{30}{72} = 12 \cdot \frac{5}{12} = 5
\]
- Number of red marbles:
\[
R = 12 \cdot P(\text{Red}) = 12 \cdot \frac{6}{72} = 12 \cdot \frac{1}{12} = 1
\]
- Number of yellow marbles:
\[
Y = 12 \cdot P(\text{Yellow}) = 12 \cdot \frac{12}{72} = 12 \cdot \frac{1}{6} = 2
\]
- Number of blue marbles:
\[
B = 12 \cdot P(\text{Blue}) = 12 \cdot \frac{24}{72} = 12 \cdot \frac{1}{3} = 4
\]
Now summarizing the number of each color:
- Number of green marbles: **5**
- Number of red marbles: **1**
- Number of yellow marbles: **2**
- Number of blue marbles: **4**
Thus, the final answer is:
Number of green marbles: **5**
Number of red marbles: **1**
Number of yellow marbles: **2**
Number of blue marbles: **4**