Answer :
Let's analyze the given scenario and determine the truth values of the statements provided:
1. Total number of marbles in the bag:
- Red marbles: 3
- Blue marbles: 5
- Violet marbles: 2
- Total marbles: \(3 + 5 + 2 = 10\)
2. Experimental probabilities:
- The frequency of choosing a blue marble is 5 out of 10 draws.
- Experimental probability of drawing blue:
[tex]\[ \text{Probability} = \frac{5}{5 + 1 + 4} = \frac{5}{10} = 0.5 \][/tex]
- The frequency of choosing a red marble is 1 out of 10 draws.
- Experimental probability of drawing red:
[tex]\[ \text{Probability} = \frac{1}{5 + 1 + 4} = \frac{1}{10} = 0.1 \][/tex]
- The frequency of choosing a violet marble is 4 out of 10 draws.
- Experimental probability of drawing violet:
[tex]\[ \text{Probability} = \frac{4}{5 + 1 + 4} = \frac{4}{10} = 0.4 \][/tex]
3. Theoretical probabilities:
- The probability of drawing a blue marble:
[tex]\[ \text{Probability} = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{5}{10} = 0.5 \][/tex]
- The probability of drawing a red marble:
[tex]\[ \text{Probability} = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]
- The probability of drawing a violet marble:
[tex]\[ \text{Probability} = \frac{\text{Number of violet marbles}}{\text{Total number of marbles}} = \frac{2}{10} = 0.2 \][/tex]
Now, let's evaluate the statements:
Statement A: The experimental probability of drawing blue is \( \frac{1}{2} \).
- From the experimental probability calculation, we found it to be 0.5. This is equivalent to \( \frac{1}{2} \).
- True
Statement B: The experimental probability of drawing red is 1.
- The experimental probability of drawing red was calculated to be 0.1.
- False
Statement C: The theoretical probability of drawing red is \( \frac{1}{10} \).
- The theoretical probability of drawing red was calculated to be 0.3.
- False
Statement D: The theoretical probability of drawing violet is 0.
- The theoretical probability of drawing violet was calculated to be 0.2.
- False
Based on the analysis, the true statement is A.
1. Total number of marbles in the bag:
- Red marbles: 3
- Blue marbles: 5
- Violet marbles: 2
- Total marbles: \(3 + 5 + 2 = 10\)
2. Experimental probabilities:
- The frequency of choosing a blue marble is 5 out of 10 draws.
- Experimental probability of drawing blue:
[tex]\[ \text{Probability} = \frac{5}{5 + 1 + 4} = \frac{5}{10} = 0.5 \][/tex]
- The frequency of choosing a red marble is 1 out of 10 draws.
- Experimental probability of drawing red:
[tex]\[ \text{Probability} = \frac{1}{5 + 1 + 4} = \frac{1}{10} = 0.1 \][/tex]
- The frequency of choosing a violet marble is 4 out of 10 draws.
- Experimental probability of drawing violet:
[tex]\[ \text{Probability} = \frac{4}{5 + 1 + 4} = \frac{4}{10} = 0.4 \][/tex]
3. Theoretical probabilities:
- The probability of drawing a blue marble:
[tex]\[ \text{Probability} = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{5}{10} = 0.5 \][/tex]
- The probability of drawing a red marble:
[tex]\[ \text{Probability} = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]
- The probability of drawing a violet marble:
[tex]\[ \text{Probability} = \frac{\text{Number of violet marbles}}{\text{Total number of marbles}} = \frac{2}{10} = 0.2 \][/tex]
Now, let's evaluate the statements:
Statement A: The experimental probability of drawing blue is \( \frac{1}{2} \).
- From the experimental probability calculation, we found it to be 0.5. This is equivalent to \( \frac{1}{2} \).
- True
Statement B: The experimental probability of drawing red is 1.
- The experimental probability of drawing red was calculated to be 0.1.
- False
Statement C: The theoretical probability of drawing red is \( \frac{1}{10} \).
- The theoretical probability of drawing red was calculated to be 0.3.
- False
Statement D: The theoretical probability of drawing violet is 0.
- The theoretical probability of drawing violet was calculated to be 0.2.
- False
Based on the analysis, the true statement is A.
