To solve a problem involving comparing theoretical and experimental probabilities of spinning a spinner, follow these steps:

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1. Theoretical Probability

Theoretical probability is calculated by assuming each outcome is equally likely. For a spinner divided into 4 sections (red, blue, yellow, green):

$P(\text{color}) = \frac{1}{4} = 0.25 \text{ or } 25\%.$

This means each color should theoretically occur $25\%$ of the time.

If the spinner is spun 20 times: $\text{Theoretical Frequency for each color} = P(\text{color}) \times 20 = 0.25 \times 20 = 5.$

So, theoretically, each color is expected to appear 5 times.

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2. Conduct the Experiment

• Spin the spinner 20 times and record the results. For instance, you might get something like:
  • Red: 6 times
  • Blue: 4 times
  • Yellow: 7 times
  • Green: 3 times

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3. Experimental Probability

Experimental probability is calculated based on actual outcomes from the experiment: $P(\text{color}) = \frac{\text{Number of times the color appears}}{\text{Total number of spins}}.$

For example, using the results above:

• $P(\text{Red}) = \frac{6}{20} = 0.3 \text{ or } 30\%$,
• $P(\text{Blue}) = \frac{4}{20} = 0.2 \text{ or } 20\%$,
• $P(\text{Yellow}) = \frac{7}{20} = 0.35 \text{ or } 35\%$,
• $P(\text{Green}) = \frac{3}{20} = 0.15 \text{ or } 15\%$.

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4. Compare Theoretical and Experimental Probabilities

• Theoretical probabilities assume perfect fairness and predictability.
• Experimental probabilities reflect real-world variations and randomness.

Compare them by listing differences. For example:

• Red: $25\% \, (\text{theoretical}) \, \text{vs.} \, 30\% \, (\text{experimental})$,
• Blue: $25\% \, \text{vs.} \, 20\%$,
• Yellow: $25\% \, \text{vs.} \, 35\%$,
• Green: $25\% \, \text{vs.} \, 15\%$.

The differences arise because of randomness and the relatively small number of trials (20 spins).

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5. Improving Accuracy

The experimental probability becomes closer to the theoretical probability as the number of spins increases. Repeating the experiment with more spins (e.g., 100 or 1000) will reduce discrepancies.

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Summary Table Example:

Color	Theoretical Probability (%)	Experimental Probability (%)
Red	25	30
Blue	25	20
Yellow	25	35
Green	25	15

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Questions for Further Analysis

1. What factors might cause experimental probabilities to differ from theoretical ones?
2. How would increasing the number of spins affect the results?
3. What is the experimental probability if the spinner is spun 40 times and each section appears 10 times?
4. Can we assume the spinner is fair based on experimental probabilities? Why or why not?
5. How do experimental probabilities relate to the Law of Large Numbers?

Tip:

When comparing probabilities, always use percentages or decimals consistently for clarity!