Question

a die is rolled 2000 times, with the following results:
outcome | 1 | 2 | 3 | 4 | 5 | 6
frequency | 332 | 365 | 441 | 290 | 280 | 292
(1) what is the theoretical probability of the given event?
p(2)
a. option
b. option
c. 0
d. option

Explanation:

To determine the theoretical probability \( P(2) \) for a fair die, we follow these steps:

Step 1: Recall the formula for theoretical probability of a die roll

A fair die has 6 equally likely outcomes: 1, 2, 3, 4, 5, 6. The theoretical probability of an event \( E \) is given by:
\[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]

Step 2: Identify favorable and total outcomes for \( P(2) \)

• The favorable outcome for \( P(2) \) is rolling a 2, so there is 1 favorable outcome.
• The total number of possible outcomes when rolling a die is 6 (since a die has 6 faces).

Step 3: Calculate \( P(2) \)

Substitute the values into the formula:
\[
P(2) = \frac{1}{6}
\]

(Note: The table provided shows experimental results, but the question asks for the theoretical probability, which depends on the nature of a fair die, not the experimental frequencies.)

Answer:

\(\frac{1}{6}\) (assuming the options include this; if the options are as follows (inferred from typical die probability questions):
A. \(\frac{1}{6}\), B. \(\frac{1}{2}\), C. \(0\), D. \(\frac{1}{3}\)
Then the correct option is A. \(\frac{1}{6}\))