Final answer:
The statements claiming that events A and B are independent and that the conditional probabilities are equal to their individual probabilities are false. Events A (drawing a blue box with a prize) and B (drawing a green box with a prize) are dependent on each other because the occurrence of one does affect the probability of the occurrence of the other.
Step-by-step explanation:
Jason has a bag that contains 888 identically shaped boxes, with 6 blue and 2 green boxes among them. To determine the truthfulness of the given statements, we must understand the concept of independent events and conditional probability. Event A represents the occurrence of drawing a blue box with a prize, and event B represents drawing a green box with a prize.
For events A and B to be independent, the occurrence of one should not affect the probability of the occurrence of the other, which means that P(A AND B) should equal P(A)P(B). However, since the outcomes of A and B are limited by the actual content of the boxes (3 blue boxes with a prize and 1 green box with a prize), the probability of one event occurring does affect the other. Therefore, events A and B are not independent. With that understanding, here is a clarification of the given statements:
- P(A|B) != P(A) because knowing that a green box with a prize was drawn changes the probability structure for drawing a blue box with a prize, and vice-versa.
- P(B|A) != P(B) because, once again, the occurrence of A changes the possible outcomes for B.
- Statement C is false because events A and B are dependent events.
- Statement D is true as the outcome of event A does affect the outcome of event B, making them dependent on each other.