In this section we learn about adding probabilities of occasions that space disjoint, i.e., occasions that have no outcomes in common. Two occasions are disjoint if it is impossible for both to happen at the exact same time. An additional name for disjoint occasions is mutually exclusive. This ar is reasonably straightforward, so this notes will certainly be quite short.

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In the adhering to discussion, the funding letters E and also F represent feasible outcomes native an experiment, and P(E) represents the probability of see outcome E.

For disjoint events, the outcomes the E or F deserve to be noted as the outcomes that E followed by the outcomes that F. The Addition Rule for the probability that disjoint occasions is:

(E or F)=(E) + (F)

Thus us can find P (E or F) if we recognize both P (E) and P (F). This is likewise true for much more than two disjoint events. If E, F, G, are all disjoint (none of castle have any outcomes in common), then:

P (E or F or G or …) = P (E) + P (F) + P (G) + ⋯

The addition rule only uses to events that room disjoint. If 2 (or more) events are no disjoint, climate this rule must be modified due to the fact that some outcomes may be counted more than once. Because that the formula (E or F) = (E) + (F), all the outcomes that room in both E and also F will be counting twice. Thus, to compute P (E or F), these double-counted outcomes need to be subtracted (once), so the each result is just counted once.

The General enhancement Rule is:

P (E or F) = P (E) + P (F) – P (E and also F),

where P (E and also F) is the collection of outcomes in both E and F. This rule is true both for disjoint events and also for non-disjoint events, because that if two events are certainly disjoint, then P (E and also F) = 0, and also the General addition Formula merely reduces come the straightforward addition formula because that disjoint events.

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Example

When selecting a map at random out of a deck of 52 cards, what is the probability of picking a queen or a heart? Define:

E = “choosing a queen”F = “choosing a heart”

E and F room not disjoint since there is one card the is both a queen and also a heart, therefore we should use the General enhancement Rule. We recognize the adhering to probabilities utilizing the classical (counting, equally-likely outcomes) method:

P (E) = P (queen) = 4/52P (F) = P (heart) = 13/52P (E and F) = P (queen the hearts) = 1/52

Therefore,

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Finally, it is often easier to calculate the probability that something will not happen rather than determining the probability the it will happen. The complement that the occasion E is the “opposite” that E. We write the match of outcome E together Ec. The complement E^c is composed of every the outcomes that space not in that occasion E

For example, once rolling one die, if event = even number, then Ec = odd number. If occasion = 1,2, then Fc = 3, 4, 5, 6.

It must make sense that the probability of the complement Ec developing is simply 1 minus the probability that event E occurs. In formula form: