Let's solve the problem.

Problem:

A single die is rolled one time. We are asked to find the probability of rolling a number greater than 4 or less than 3.

Step 1: Understanding the Problem

A standard die has six faces numbered from 1 to 6. The total number of outcomes when rolling a die is 6.

• Rolling a number greater than 4 includes the outcomes: $5, 6$.
  The number of favorable outcomes here is $2$.

• Rolling a number less than 3 includes the outcomes: $1, 2$.
  The number of favorable outcomes here is also $2$.

Step 2: Combine the Outcomes

The problem asks for the union of these two events. Using the formula for probability of a union:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• $P(A)$: Probability of rolling a number greater than 4 = $\frac{2}{6} = \frac{1}{3}$.
• $P(B)$: Probability of rolling a number less than 3 = $\frac{2}{6} = \frac{1}{3}$.
• $P(A \cap B)$: The overlap (intersection) between the two events is $0$ because no number can be both greater than 4 and less than 3.

Thus,

$P(A \cup B) = \frac{1}{3} + \frac{1}{3} - 0 = \frac{2}{3}.$

Final Answer:

The probability is: $\boxed{\frac{2}{3}}$

Would you like me to elaborate further?

Related Questions:

1. How do we calculate the probability of complementary events?
2. What is the probability of rolling an even number on a single die roll?
3. How does the formula for probability of a union apply to overlapping events?
4. If the die had 8 faces, how would the probability change?
5. What is the probability of rolling a number greater than 2 and less than 5?

Tip:

Always check for overlaps between events when using the formula for $P(A \cup B)$. This avoids double-counting shared outcomes!