You roll two six-sided fair dice.

a.  Let A be the event that either a 4 or 5 is rolled first followed by an even number.

Let's break down the problem step by step:

### Understanding Event A
Event \( A \) is defined as rolling either a 4 or 5 first, followed by an even number on the second roll.

### Possible Outcomes
1. **First Roll: 4 or 5**  
   There are two possible outcomes for the first roll:
   - Roll a 4
   - Roll a 5

2. **Second Roll: Even Number**  
   The even numbers on a six-sided die are 2, 4, and 6. So, the possible outcomes for the second roll are:
   - Roll a 2
   - Roll a 4
   - Roll a 6

### Probability Calculation
The probability of event \( A \) happening is the product of the probabilities of each step occurring in sequence.

1. **Probability of rolling a 4 or 5 on the first roll:**  
   There are 2 favorable outcomes (rolling a 4 or 5) out of 6 possible outcomes:
   \[
   P(\text{first roll is 4 or 5}) = \frac{2}{6} = \frac{1}{3}
   \]

2. **Probability of rolling an even number on the second roll:**  
   There are 3 favorable outcomes (rolling a 2, 4, or 6) out of 6 possible outcomes:
   \[
   P(\text{second roll is even}) = \frac{3}{6} = \frac{1}{2}
   \]

### Probability of Event A
The probability of event \( A \) is the product of the two independent probabilities:
\[
P(A) = P(\text{first roll is 4 or 5}) \times P(\text{second roll is even}) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}
\]

### Final Answer
The probability of event \( A \) (rolling either a 4 or 5 first, followed by an even number) is \( \frac{1}{6} \).

Would you like more details on any part of this, or do you have any further questions?

Here are five related questions to explore further:
1. What is the probability of rolling two even numbers consecutively?
2. What is the probability of rolling two 6s in a row?
3. What is the probability of rolling a 4 followed by a 5?
4. What is the probability of not rolling a 4 or 5 on the first roll?
5. How would the probability change if a die had more sides, like an 8-sided die?

**Tip:** When calculating the probability of sequential events, multiply the probabilities of each individual event, assuming they are independent.

Explore the probability of rolling either a 4 or 5 first followed by an even number on two fair six-sided dice. Detailed step-by-step calculation and explanation provided. Suitable for high school students studying probability and dice rolling.

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