To assess Sean's claim about the theoretical probability of rolling a number on a fair six-sided die (or number cube), we should first clarify what the theoretical probabilities are for each number.
In a fair six-sided die, each face (1 through 6) has an equal probability of landing face up when rolled. Therefore, the theoretical probability \( P \) of rolling any specific number (including 1) is:
\[ P(\text{rolling a specific number}) = \frac{1}{6} \]
This means that the theoretical probability of rolling a 1 is \( \frac{1}{6} \), which is the same as the probability of rolling a 2, 3, 4, 5, or 6.
Now, let's analyze Sean's observation of the results from the rolls:
- Sean recorded the following rolls: 1, 6, 2, 1, 5, 4, 2, 2, 1, 4, 5, 1, 3, 4, 1.
- In this sequence, the number 1 appeared 5 times out of 15 total rolls.
To calculate the experimental probability of rolling a 1 based on Sean's results:
\[ P(\text{rolling a 1}) = \frac{\text{Number of times 1 is rolled}}{\text{Total number of rolls}} = \frac{5}{15} = \frac{1}{3} \]
Now, let's also compute the experimental probabilities for the other numbers based on his data:
- Number 2: rolled 3 times → \( P(2) = \frac{3}{15} = \frac{1}{5} \)
- Number 3: rolled 1 time → \( P(3) = \frac{1}{15} \)
- Number 4: rolled 3 times → \( P(4) = \frac{3}{15} = \frac{1}{5} \)
- Number 5: rolled 2 times → \( P(5) = \frac{2}{15} \)
- Number 6: rolled 1 time → \( P(6) = \frac{1}{15} \)
Comparing these experimental probabilities reveals that the experimental probability of rolling a 1 (\(\frac{1}{3}\)) is higher than that of rolling 2, 3, 4, 5, or 6. However, Sean's statement about theoretical probabilities is incorrect; theoretically, all numbers have an equal chance of \(\frac{1}{6}\).
Sean's error lies in misunderstanding the difference between theoretical probability (which states that each number has a probability of \(\frac{1}{6}\)) and experimental probability, which is based on the actual outcomes of the rolls. Just because in this specific instance 1 appeared more frequently does not mean it has a greater theoretical probability; the theoretical probabilities remain equal for all numbers when considering a fair die.
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