For any events, A and B, the following properties always hold.

Probabilities are always between 0 and 1

0  ≤  P(A)  ≤  1

Meaning of probabilities 0 and 1

If the event A cannot happen then P(A)  =  0

If the event A is certain to happen then P(A)  =  1

Probability that an event does not happen

P(A does not happen)  =  1 - P(A)

Addition law

When two events cannot happen together, they are said to be mutually exclusive. If A and B are mutually exclusive,

P(A or B)  =  P(A) + P(B)

If the events A and B are not mutually exclusive,

P(A or B)  <  P(A) + P(B)

Independence

When sampling with replacement from a finite population, the choice of each value does not depend on the values previously selected. The successive values are then called independent. This also holds when sampling from an infinite population (distribution).

On the other hand, if sampling without replacement from a finite population, successive sample values are not independent since the second value selected cannot be the same as the first value, so knowing the first value affects the probabilities when the second value is selected.