The probability of rolling a 4 on a single throw of a fair 6-sided die is 1/6, since there is only one way to roll a 4 and there are 6 equally likely outcomes in total.
The probability of rolling an even number on a single throw of a fair 6-sided die is 3/6, or 1/2, since there are three even numbers (2, 4, and 6) out of six possible outcomes.
To find the probability of rolling a 4 or an even number on a single throw, we can add the probabilities of these two events:
P(4 or even) = P(4) + P(even) - P(4 and even)
where P(4 and even) is the probability of rolling a 4 and an even number on the same throw. Since there is only one outcome (rolling a 4), which is not even, this probability is 0.
Therefore:
P(4 or even) = P(4) + P(even) - P(4 and even)
= 1/6 + 1/2 - 0
= 2/3
So the probability of rolling a 4 or an even number on a single throw of a 6-sided die is 2/3.
If the die is thrown 2 times, the probability of rolling a 4 or an even number on both throws is the product of the probabilities of rolling a 4 or an even number on each throw:
P(4 or even on both throws) = P(4 or even) × P(4 or even)
= (2/3) × (2/3)
= 4/9
Therefore, the probability of rolling a 4 or an even number on both throws of a 6-sided die is 4/9.