Mutually Exclusive, Exhaustive and Independent Events
Mutually exclusive events are events that cannot happen simultaneously, or such that the occurrence of one means that the other cannot subsequently occur. if you throw a dice, only one face can be on top, so the events which described by the uppermost face are mutually exclusive.
The probabilities of mutually exclusive events can be added to find the overall probability of one of the events happening, so that if
and
are mutually exclusive events, then
This can be extended to events
in the obvious way.
Mutually exhaustive events are a complete set of possible outcomes, so that one of these events must happen. If you throw a dice, the face which appears uppermost must be one of 1, 2, 3, 4, 5 or 6, so so the events which described by the uppermost face are mutually exhaustive.
Independent events are events that do not depend on each other in any way whatsoever. Successive scores on a fair dice repeatedly thrown is a good example. The score is a random event, and the score on each throw is independent of preceding or succeeding scores.
Independent events are events that exert no influence on each other – the are mutually independent so that ifis independent of
thenis independent of
Independent events obey a multiplication law. The probability of independent eventsandboth happening is equal to the product of the probability of
with the probability of
This can be written
This can be extended to eventsin the obvious way.
It is important to note that ifandare independent, then
since the probability ofandboth happening is not taken into account here.