Final answer:
Two events are considered independent in probability theory if the occurrence of one does not affect the probability of the other occurring. This is indicated by the product of the probabilities of the two individual events being equal to the probability of both events occurring together. For example, if P(G) = 0.6 and P(H) = 0.5 are independent as P(G AND H) = P(G)P(H) = 0.3.
Step-by-step explanation:
In mathematics, particularly in probability theory, we say that two events are independent if the occurrence of one event does not impact the likelihood of the other event occurring. The mathematical condition for two events, A and B, to be independent is given by the equation P(A AND B) = P(A)P(B), where P represents the probability.
For example, suppose we have two events G (taking a math class) and H (taking a science class) with their respective probabilities P(G) = 0.6 and P(H) = 0.5. If the probability of both events happening together is P(G AND H) = 0.3, then to determine if events G and H are independent, we compute P(G)*P(H) and check if it equals P(G AND H). Indeed, 0.6*0.5 = 0.3, which matches the value of P(G AND H), thus confirming that events G and H are independent.
Similarly, when you are working on problems involving these concepts, you would need to gather the probabilities of each event and the probability of them occurring together. When you have these values, simply plug them into the equation and see if they match up to determine independence.