Final answer:
The probability of getting heads and an even number on an eight-sided die is 0.25. The probability of getting tails and a prime number less than 4 (which is three), is 0.0625. Events A (heads and an even number) and B (heads and a three) are mutually exclusive with P(A AND B) = 0.
Step-by-step explanation:
The probability that Oscar gets heads and an even number on an eight-sided die is determined by the outcomes that satisfy both conditions. Since a coin has two sides, the probability of getting heads (H) is 0.5. For an eight-sided die, the even numbers are 2, 4, 6, and 8, giving us four favorable outcomes. As each face of the die is equally likely, the probability of getting an even number is 4 out of 8, or 0.5. Thus, the probability of both heads and an even number is 0.5 (from the coin) multiplied by 0.5 (from the die), which gives us a combined probability of 0.25 or 25%.
For the second part, the probability that Oscar gets tails and a prime number less than 4 is again 0.5 for getting tails. The prime numbers less than 4 on an eight-sided die are 2 and 3, but since 2 is even, the only prime number less than 4 that we are considering is 3. There is only one such outcome on the die, so the probability of getting a three is 1 out of 8, or 0.125. The combined probability of tails and a prime number less than 4 is 0.5 (from the coin) times 0.125 (from the die), resulting in a probability of 0.0625 or 6.25%.
Events A (heads and an even number) and B (heads and a three) are mutually exclusive because they cannot both occur at the same time. In other words, the die cannot land on an even number and the number three at the same time. Hence, P(A AND B) = 0, confirming that events A and B are mutually exclusive.