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.

Question

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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.

Answer 1

To answer the questions, we will work systematically through the experimental and theoretical probabilities associated with the spinner.

A. Experimental Probabilities

Data from Trials:

Let's assume the recorded outcomes for each trial of 10 spins per trial are:

• Trial 1: A: 3, B: 4, C: 3
• Trial 2: A: 5, B: 2, C: 3
• Trial 3: A: 2, B: 5, C: 3

Total Counts: Now, we will sum the counts of spins for each letter across the three trials:

• Total for A: 3 + 5 + 2 = 10
• Total for B: 4 + 2 + 5 = 11
• Total for C: 3 + 3 + 3 = 9

Total Spins: Total number of spins = 10 spins/trial × 3 trials = 30 spins

Calculating Experimental Probabilities:

• Probability of A (P(A)) = (Total A spins) / (Total spins) = 10 / 30 = 1/3 ≈ 0.33
• Probability of B (P(B)) = (Total B spins) / (Total spins) = 11 / 30 ≈ 0.367
• Probability of C (P(C)) = (Total C spins) / (Total spins) = 9 / 30 = 3/10 = 0.30

B. Theoretical Probabilities

For a spinner divided into three equal regions:

• Each section (A, B, C) has an equal chance of landing on because the regions are equal in size. Therefore, the theoretical probability for each outcome is:
• P(A) = 1/3
• P(B) = 1/3
• P(C) = 1/3

C. Comparison of Experimental and Theoretical Probabilities:

Experimental Probabilities:

• P(A) ≈ 0.33
• P(B) ≈ 0.367
• P(C) ≈ 0.30

Theoretical Probabilities:

• P(A) = 0.33
• P(B) = 0.33
• P(C) = 0.33

Agreement: The experimental probabilities do not exactly match the theoretical probabilities. A reason for this discrepancy could be due to:

1. Sample Size: The small sample size (only 30 spins) can lead to variation in outcomes and does not perfectly represent the ideal theoretical probabilities.
2. Random Variability: Each trial's outcome is subject to randomness, meaning that actual results will likely vary from expected results, especially when the sample size is small.
3. Systematic Bias: If there were any issues with the spinner (e.g., not spinning evenly), that could cause an unequal distribution of outcomes.

In summary, while the experimental probabilities derived from the trials are close to the theoretical probabilities, they do not agree exactly due to the limitations of sampling and random variation inherent in probability experiments. As the number of spins increases, the experimental probabilities are expected to converge closer to their theoretical values.