Final answer:
The correct answer to the probability question is that G and H are independent events since P(G|H) = P(G). The differences between P(G₁ AND R₂) and P(R₁|G₂) represent joint probability and conditional probability, respectively. Events are mutually exclusive if they cannot occur at the same time, and independent if the occurrence of one does not affect the probability of the other occurring.
Step-by-step explanation:
Understanding Probability and Logical Reasoning
When considering the statement P(G|H) = P(G), this indicates that the probability of G occurring given that H has occurred is the same as the probability of G occurring regardless of H. Therefore, the correct answer is D. G and H are independent events. Independence in probability means the occurrence of one event does not affect the probability of the occurrence of another event.
In the context of P(G₁ AND R₂) versus P(R₁|G₂), the former represents the probability that both events G₁ and R₂ occur simultaneously, while the latter specifies the probability of R₁ occurring given that G₂ has occurred. These notations reflect different scenarios and different sample spaces in probability.
The concept of mutually exclusive events is relevant when discussing whether events F and G cannot occur at the same time. If events, like J and H, cannot occur together, they are mutually exclusive events. Without additional information, we cannot determine mutual exclusivity just by event labels.
Independence between events A and B can be determined by assessing if P(A AND B) = P(A)P(B). If this equation holds true, A and B are indeed independent events; if not, they are dependent on each other.