Question
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Question
Ms. Lew has a bag of 3D shapes she uses to teach her 3rd graders. The bag contains 6 cubes, 12 spheres, and 8 pyramids. She records the shape she pulls out of the bag for 20 trials, replacing the shape each time. The following table summarizes her results.
Shape Frequency
Cube 5
Sphere 8
Pyramid 7
Which of the following statements about experimental and theoretical probabilities is correct?
Responses
A The theoretical probability of pulling a cube is equal to the experimental probability.The theoretical probability of pulling a cube is equal to the experimental probability.
B The theoretical probability of pulling a sphere is greater than the experimental probability.The theoretical probability of pulling a sphere is greater than the experimental probability.
C The theoretical probability of pulling a sphere is less than the experimental probability.The theoretical probability of pulling a sphere is less than the experimental probability.
D The theoretical probability of pulling a pyramid is greater than the experimental probability.
Answers
GPT-4o mini
To determine the correct statement about the experimental and theoretical probabilities for Ms. Lew's shapes, we first need to calculate the theoretical probabilities of each shape based on the total shapes in the bag and then compare them to the experimental probabilities derived from the trials.
1. **Total Shapes**:
- Cubes: 6
- Spheres: 12
- Pyramids: 8
- Total = 6 + 12 + 8 = 26 shapes
2. **Theoretical Probabilities**:
- Probability of pulling a cube:
\[
P(\text{Cube}) = \frac{\text{Number of Cubes}}{\text{Total Shapes}} = \frac{6}{26} = \frac{3}{13} \approx 0.2308
\]
- Probability of pulling a sphere:
\[
P(\text{Sphere}) = \frac{\text{Number of Spheres}}{\text{Total Shapes}} = \frac{12}{26} = \frac{6}{13} \approx 0.4615
\]
- Probability of pulling a pyramid:
\[
P(\text{Pyramid}) = \frac{\text{Number of Pyramids}}{\text{Total Shapes}} = \frac{8}{26} = \frac{4}{13} \approx 0.3077
\]
3. **Experimental Probabilities** from the trials:
- Experimental probability of pulling a cube:
\[
P(\text{Cube}) = \frac{5}{20} = 0.25
\]
- Experimental probability of pulling a sphere:
\[
P(\text{Sphere}) = \frac{8}{20} = 0.4
\]
- Experimental probability of pulling a pyramid:
\[
P(\text{Pyramid}) = \frac{7}{20} = 0.35
\]
4. **Comparing Theoretical and Experimental Probabilities**:
- For cubes:
- Theoretical: \(0.2308\)
- Experimental: \(0.25\)
- Experimental probability is greater.
- For spheres:
- Theoretical: \(0.4615\)
- Experimental: \(0.4\)
- Theoretical probability is greater.
- For pyramids:
- Theoretical: \(0.3077\)
- Experimental: \(0.35\)
- Experimental probability is greater.
Now, let's evaluate the statements:
- **A.** The theoretical probability of pulling a cube is equal to the experimental probability. (Incorrect)
- **B.** The theoretical probability of pulling a sphere is greater than the experimental probability. (Correct)
- **C.** The theoretical probability of pulling a sphere is less than the experimental probability. (Incorrect)
- **D.** The theoretical probability of pulling a pyramid is greater than the experimental probability. (Incorrect)
Thus, the correct answer is **B**: The theoretical probability of pulling a sphere is greater than the experimental probability.
1. **Total Shapes**:
- Cubes: 6
- Spheres: 12
- Pyramids: 8
- Total = 6 + 12 + 8 = 26 shapes
2. **Theoretical Probabilities**:
- Probability of pulling a cube:
\[
P(\text{Cube}) = \frac{\text{Number of Cubes}}{\text{Total Shapes}} = \frac{6}{26} = \frac{3}{13} \approx 0.2308
\]
- Probability of pulling a sphere:
\[
P(\text{Sphere}) = \frac{\text{Number of Spheres}}{\text{Total Shapes}} = \frac{12}{26} = \frac{6}{13} \approx 0.4615
\]
- Probability of pulling a pyramid:
\[
P(\text{Pyramid}) = \frac{\text{Number of Pyramids}}{\text{Total Shapes}} = \frac{8}{26} = \frac{4}{13} \approx 0.3077
\]
3. **Experimental Probabilities** from the trials:
- Experimental probability of pulling a cube:
\[
P(\text{Cube}) = \frac{5}{20} = 0.25
\]
- Experimental probability of pulling a sphere:
\[
P(\text{Sphere}) = \frac{8}{20} = 0.4
\]
- Experimental probability of pulling a pyramid:
\[
P(\text{Pyramid}) = \frac{7}{20} = 0.35
\]
4. **Comparing Theoretical and Experimental Probabilities**:
- For cubes:
- Theoretical: \(0.2308\)
- Experimental: \(0.25\)
- Experimental probability is greater.
- For spheres:
- Theoretical: \(0.4615\)
- Experimental: \(0.4\)
- Theoretical probability is greater.
- For pyramids:
- Theoretical: \(0.3077\)
- Experimental: \(0.35\)
- Experimental probability is greater.
Now, let's evaluate the statements:
- **A.** The theoretical probability of pulling a cube is equal to the experimental probability. (Incorrect)
- **B.** The theoretical probability of pulling a sphere is greater than the experimental probability. (Correct)
- **C.** The theoretical probability of pulling a sphere is less than the experimental probability. (Incorrect)
- **D.** The theoretical probability of pulling a pyramid is greater than the experimental probability. (Incorrect)
Thus, the correct answer is **B**: The theoretical probability of pulling a sphere is greater than the experimental probability.