What Is Mutually Inclusive?

Spades or a king? Why not choose both?
••• playing cards image by Mijakowska from Fotolia.com

Whereas a mutually exclusive event is one wherein two events cannot happen at the same time (getting heads and tails in a single coin toss), a mutually inclusive event allows both events to occur in a single trial (drawing a spade and a king).


The main draw of a mutually inclusive event is that it allows two different events to occur simultaneously. Due to this, be aware that if one event occurs, it does not necessarily preclude another event occurring at the same time.


Drawing a black card or a king serves as an example of a mutually inclusive event. The odds of drawing a black card are 26 out of 52, and the odds of drawing a king are 4 out of 52. However, because drawing either a black card or a king is considered a success, the true probability of this event would be 28 out of 52, because half the deck is black (26 out of 52) and the drawer has the added advantage of the two extra red king cards (26 out of 52 plus 2 out of 52 equals 28 out of 52).

Generalized, the equation of mutually inclusive events can be written as: P(a or b) = P(a) + P(b) - P(a and b)


The math behind mutually inclusive events is used in most instances where probabilities arise and can occur simultaneously. As such, the equation cannot be applied to dependent variables, wherein one event depends on another happening. For example, to calculate the probability of drawing a black card or a king twice in a row, the same equation used with a mutually inclusive event cannot be used, because the two cards cannot be drawn at the same time. Furthermore, the probability for the second card will be changed because there is one less card in the deck.

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