Final answer:
The two important properties of probabilities are independence and mutual exclusivity. Mutually exclusive events cannot occur together, evidenced by the probability of their joint occurrence being zero. Independent events do not affect each other's probability of occurring.
Step-by-step explanation:
Important Properties of Probabilities
The two important properties of probabilities that are essential in the field of statistics are independence and mutual exclusivity. These properties are used to determine the relationship between two events, which can consequently affect calculations involving their probabilities.
Mutually Exclusive Events
Mutually exclusive events are those events that cannot happen at the same time. The probability of both occurring together is zero (P(A AND B) = 0). For example, when flipping a coin, the events 'Heads' and 'Tails' are mutually exclusive.
Independent Events
Independent events are those where the occurrence of one event does not affect the probability of occurrence of another event. In terms of probability, P(A AND B) = P(A)P(B), and P(B|A) = P(B). An example of independent events is flipping a coin and rolling a die simultaneously — the result of one does not impact the result of the other.
By understanding these concepts, one can apply the multiplication rule and the addition rule correctly. The multiplication rule is used for finding the probability of both events occurring (P(A AND B)), and the addition rule is used for computing the probability of either event occurring (P(A OR B)).