1. If all possible outcomes of two events are different, we say that the two events are mutually exclusive or incompatible. The two events of "drawing spades" and "drawing clubs" are incompatible, but the two events of "drawing spades" and "drawing a" are not, because "drawing spades" belong to these two events. When two events are mutually exclusive, the total number of results of either event is the sum of their respective results, so we have a simple conclusion:

When two events are mutually exclusive, the probability of at least one event is the sum of their respective probabilities.

This is the addition theorem of probability.

2. For the problem of 10 ball, the probability of two events occurring at the same time also appears in the form of product, in which the first multiplier is the probability of one event (smaller number) and the second multiplier is the conditional probability of getting a green number when getting a smaller number. So the two calculations are the same in form, the only difference is whether the result of the first event will affect the second event. In these two calculations, we use the multiplication theorem of probability:

The probability of two events occurring at the same time is the product of the probability of the first event and the probability of the second event when the first event occurs.

3. Any event will or will not happen. The total probability is divided into two parts: event occurrence and non-occurrence. So if we can find the probability that the event will not happen, and subtract this probability from 100%, we can infer the probability that it will happen. This is a clever trick.

The core creed of decision theory: reasonable decision can maximize the average utility of the result. You can never be sure that your actions will bring the best results, but you have made full use of the information you have.

You can't ask for more.