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Use the table to answer the question.

The table presents the frequency of occurrences for numbers 2, 4, 6, 8, and 10 when Jade tossed a cube.
Outcomes 2 4 6 8 10 12
Frequency 10 9 6 15 13 8
Jede tossed a cube with faces numbered with 2, 4, 6, 8, 10, and 12. The results are recorded in the table. What is the smallest discrepancy between the experimental and the expected probability of this experiment? Write your answer in 3 decimal places, rounded to the nearest thousandth.

1 answer

To find the smallest discrepancy between the experimental and expected probability for Jade's experiment of tossing a cube with faces numbered 2, 4, 6, 8, 10, and 12, we first need to calculate the experimental and expected probabilities for each outcome.

  1. Calculate total frequency: The total number of outcomes (frequency) is calculated by summing the frequencies of all outcomes: \[ \text{Total Frequency} = 10 + 9 + 6 + 15 + 13 + 8 = 61 \]

  2. Calculate experimental probabilities: Next, we calculate the experimental probability for each number by dividing its frequency by the total frequency:

    • For 2: \[ P(2) = \frac{10}{61} \approx 0.16393442623 \]
    • For 4: \[ P(4) = \frac{9}{61} \approx 0.14754098361 \]
    • For 6: \[ P(6) = \frac{6}{61} \approx 0.09836065574 \]
    • For 8: \[ P(8) = \frac{15}{61} \approx 0.24590163934 \]
    • For 10: \[ P(10) = \frac{13}{61} \approx 0.2131147541 \]
    • For 12: \[ P(12) = \frac{8}{61} \approx 0.13114754098 \]

  3. Calculate the expected probabilities: Since the cube is fair, each face (2, 4, 6, 8, 10, 12) has an equal probability of \( \frac{1}{6} \): \[ P(\text{expected}) = \frac{1}{6} \approx 0.16666666667 \]

  4. Calculate discrepancies for each outcome: Next, we'll calculate the discrepancy between the experimental and expected probabilities for each outcome:

    • For 2: \[ |P(2) - P(\text{expected})| \approx |0.16393442623 - 0.16666666667| \approx 0.00273224044 \]
    • For 4: \[ |P(4) - P(\text{expected})| \approx |0.14754098361 - 0.16666666667| \approx 0.01912568306 \]
    • For 6: \[ |P(6) - P(\text{expected})| \approx |0.09836065574 - 0.16666666667| \approx 0.06830601093 \]
    • For 8: \[ |P(8) - P(\text{expected})| \approx |0.24590163934 - 0.16666666667| \approx 0.07923497267 \]
    • For 10: \[ |P(10) - P(\text{expected})| \approx |0.2131147541 - 0.16666666667| \approx 0.04644808743 \]
    • For 12: \[ |P(12) - P(\text{expected})| \approx |0.13114754098 - 0.16666666667| \approx 0.03551912569 \]

  5. Find the smallest discrepancy: Now, we find the smallest discrepancy among all calculated discrepancies: \[ \text{Smallest Discrepancy} = \min(0.00273224044, 0.01912568306, 0.06830601093, 0.07923497267, 0.04644808743, 0.03551912569) \approx 0.00273224044 \]

  6. Round to three decimal places: Finally, rounding the smallest discrepancy to three decimal places, we get: \[ \text{Smallest Discrepancy} \approx 0.003 \]

Thus, the smallest discrepancy between the experimental and expected probability is 0.003.