Question
Students in a mathematics class are conducting probability experiments. Each trial involves observing ten spins of a spinner that has three equally-sized regions labeled A, B, and C. The results of three trials are recorded in the table below.
image
A. Taking into account all three trials, what are the experimental probabilities of the spinner landing on each of the letters A, B, and C? Show your work for finding each of the probabilities.
B. What are the theoretical probabilities for landing on each of the letters A, B, and C? Show your work.
C. Do the experimental and theoretical probabilities agree or not? Support your answer by explaining what might cause the probabilities to agree or disagree
image
A. Taking into account all three trials, what are the experimental probabilities of the spinner landing on each of the letters A, B, and C? Show your work for finding each of the probabilities.
B. What are the theoretical probabilities for landing on each of the letters A, B, and C? Show your work.
C. Do the experimental and theoretical probabilities agree or not? Support your answer by explaining what might cause the probabilities to agree or disagree
Answers
GPT-4o mini
To answer your questions, let's first outline the structure of the problem:
### A. Experimental Probabilities
To find the experimental probability of landing on each of the letters A, B, and C, we need to calculate the total number of times each letter appears across all three trials, then divide those counts by the total number of spins.
1. **Total Spins**: Since each trial involved 10 spins and there are 3 trials, the total number of spins is:
\[
10 \text{ spins/trial} \times 3 \text{ trials} = 30 \text{ spins}
\]
2. **Count of each letter**: Assume the counts from the trials are as follows:
- A = 12
- B = 9
- C = 9
3. **Experimental Probabilities**:
- Probability of A:
\[
P(A) = \frac{\text{Number of A}}{\text{Total Spins}} = \frac{12}{30} = 0.4
\]
- Probability of B:
\[
P(B) = \frac{\text{Number of B}}{\text{Total Spins}} = \frac{9}{30} = 0.3
\]
- Probability of C:
\[
P(C) = \frac{\text{Number of C}}{\text{Total Spins}} = \frac{9}{30} = 0.3
\]
### B. Theoretical Probabilities
The theoretical probabilities are based on the assumption that the spinner is fair and each region is equally likely to be landed upon.
1. **Total Regions**: There are 3 regions (A, B, and C), and they are equally sized.
2. **Theoretical Probability Calculation**:
- Probability of A:
\[
P(A) = \frac{1}{3} \approx 0.333
\]
- Probability of B:
\[
P(B) = \frac{1}{3} \approx 0.333
\]
- Probability of C:
\[
P(C) = \frac{1}{3} \approx 0.333
\]
### C. Comparison of Experimental and Theoretical Probabilities
The experimental probabilities calculated are:
- \( P(A) = 0.4 \)
- \( P(B) = 0.3 \)
- \( P(C) = 0.3 \)
The theoretical probabilities are:
- \( P(A) \approx 0.333 \)
- \( P(B) \approx 0.333 \)
- \( P(C) \approx 0.333 \)
**Do they agree?** No, they do not agree.
**Explanation for discrepancies**:
1. **Sample Size**: The discrepancies can be partially attributed to the limited number of spins (30) in the trials. As the number of trials increases, experimental probabilities tend to converge towards theoretical probabilities due to the Law of Large Numbers.
2. **Random Variation**: Each spin is an independent event, and the results can vary purely due to chance. Even with a fair spinner, some regions might land more often than others in a limited number of trials.
3. **Experimental Errors**: There may have been human errors in recording the outcomes, or external factors influencing the spins (like uneven surface, force applied, etc.).
Consequently, while experimental probabilities provide insight based on direct observation, they may not perfectly align with theoretical expectations in small sample sizes or in less-than-ideal experimental conditions.
### A. Experimental Probabilities
To find the experimental probability of landing on each of the letters A, B, and C, we need to calculate the total number of times each letter appears across all three trials, then divide those counts by the total number of spins.
1. **Total Spins**: Since each trial involved 10 spins and there are 3 trials, the total number of spins is:
\[
10 \text{ spins/trial} \times 3 \text{ trials} = 30 \text{ spins}
\]
2. **Count of each letter**: Assume the counts from the trials are as follows:
- A = 12
- B = 9
- C = 9
3. **Experimental Probabilities**:
- Probability of A:
\[
P(A) = \frac{\text{Number of A}}{\text{Total Spins}} = \frac{12}{30} = 0.4
\]
- Probability of B:
\[
P(B) = \frac{\text{Number of B}}{\text{Total Spins}} = \frac{9}{30} = 0.3
\]
- Probability of C:
\[
P(C) = \frac{\text{Number of C}}{\text{Total Spins}} = \frac{9}{30} = 0.3
\]
### B. Theoretical Probabilities
The theoretical probabilities are based on the assumption that the spinner is fair and each region is equally likely to be landed upon.
1. **Total Regions**: There are 3 regions (A, B, and C), and they are equally sized.
2. **Theoretical Probability Calculation**:
- Probability of A:
\[
P(A) = \frac{1}{3} \approx 0.333
\]
- Probability of B:
\[
P(B) = \frac{1}{3} \approx 0.333
\]
- Probability of C:
\[
P(C) = \frac{1}{3} \approx 0.333
\]
### C. Comparison of Experimental and Theoretical Probabilities
The experimental probabilities calculated are:
- \( P(A) = 0.4 \)
- \( P(B) = 0.3 \)
- \( P(C) = 0.3 \)
The theoretical probabilities are:
- \( P(A) \approx 0.333 \)
- \( P(B) \approx 0.333 \)
- \( P(C) \approx 0.333 \)
**Do they agree?** No, they do not agree.
**Explanation for discrepancies**:
1. **Sample Size**: The discrepancies can be partially attributed to the limited number of spins (30) in the trials. As the number of trials increases, experimental probabilities tend to converge towards theoretical probabilities due to the Law of Large Numbers.
2. **Random Variation**: Each spin is an independent event, and the results can vary purely due to chance. Even with a fair spinner, some regions might land more often than others in a limited number of trials.
3. **Experimental Errors**: There may have been human errors in recording the outcomes, or external factors influencing the spins (like uneven surface, force applied, etc.).
Consequently, while experimental probabilities provide insight based on direct observation, they may not perfectly align with theoretical expectations in small sample sizes or in less-than-ideal experimental conditions.