Complementary event

In a random experiment, the probabilities of all possible events (the sample space) must total to 1— that is, some outcome must occur on every trial.

For two events to be complements, they must be collectively exhaustive, together filling the entire sample space.

Suppose one throws an ordinary six-sided die eight times.

The technique is wrong because the eight events whose probabilities got added are not mutually exclusive.

One may resolve this overlap by the principle of inclusion-exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: