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Complementary and Mutually Exclusive Events

Complementary Events
Complementary events are generally the two possible outcomes of an event in all possible ways.

Example
While flipping a coin we may get either head or the tails. There are no other options exists so these events are said to be complementary events.
On rolling a die we generally have more than 2 possibilities so they are not the mutually exclusive events.
The one outcome of an event E is happening and the other event E’ is not happening they are said to complementary.

In the above example,
E: P (Head) = ½
E’ : P( Not getting head (Tail)) = ½

Conditions for Complementary Events
The sum of the probabilities of the complementary event is generally 1.
P(E) + P(E’) = 1

The above can generally be rearranged as,
P ( E’) = 1 – P (E).

Mutually Exclusive Events
The mutually exclusive events are generally called as the disjoint events. This can be generally two or more outcomes but that cannot occur at the same time.

In other words a mutually exclusive event has no elements in common. 
If the union of the two mutually exclusive events gives the sample space itself as the result without any changes, then they are called as the exhaustive events.

Example
Picking one card from the deck of cards and choosing it as an ace or the king is generally the mutually exclusive events.

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Probability Formula – Mutually Exclusive Events
Probability of a mutually exclusive event happening = Number of ways it can happen / Total number of outcomes.

Conditions for Mutually Exclusive Events.

  1. If two events are found to be mutually exclusive events then they are impossible for them to occur together.

              P(A and B) = 0.

  1. If two events A and B are occurring then probabilities of the event A or B is the sum of the individual probabilities.

P (A or B) = P (A) + P (B).
Note: All the complementary events are mutually exclusive events but not all the mutually exclusive events are complementary events.

Reference

https://www.mathsisfun.com/data/probability-events-mutually-exclusive.html

http://artemate.altervista.org/dFileProbability/04-Complementary-Addition-RULE.ppt.pdf

https://www.shmoop.com/study-guides/math/basic-statistics-probability/complementary-mutually-exclusive-events

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