To solve this problem, let’s analyze the events $A$ and $B$ and calculate $P(A \cup B)$.

1. Event $A$: The event that the die shows an even number.

   • Possible outcomes for $A$ are: $\{2, 4, 6\}$.
   • Since a standard die has 6 faces, $P(A) = \frac{3}{6} = 0.5$.

2. Event $B$: The event that the die shows a number greater than 2.

   • Possible outcomes for $B$ are: $\{3, 4, 5, 6\}$.
   • Thus, $P(B) = \frac{4}{6} = \frac{2}{3} \approx 0.667$.

3. Intersection of $A$ and $B$ (i.e., $A \cap B$): The event that the die shows a number that is both even and greater than 2.

   • Possible outcomes for $A \cap B$ are: $\{4, 6\}$.
   • So, $P(A \cap B) = \frac{2}{6} = \frac{1}{3} \approx 0.333$.

4. Union of $A$ and $B$ (i.e., $A \cup B$): The probability that the die shows either an even number or a number greater than 2.

   • We use the formula for the probability of the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
   • Substituting the values: $P(A \cup B) = 0.5 + 0.667 - 0.333 = 0.834$

Therefore, the answer, rounded to three decimal places, is: $P(A \cup B) = 0.834$

Would you like further clarification on this solution?

Related Questions:

1. What is the probability of rolling a number less than or equal to 2?
2. How would the probability change if the die were 8-sided instead of 6-sided?
3. What is the probability of rolling a number that is odd or greater than 4?
4. If two dice are rolled, what is the probability that at least one shows an even number?
5. What is the probability of rolling a prime number on a 6-sided die?

Tip:

When calculating probabilities with unions, remember to subtract the intersection to avoid double-counting outcomes shared by both events.