3. Combining Probabilities

In this section we learn about adding probabilities of events that are disjoint, i.e., events that have no outcomes in common. Two events are disjoint if it is impossible for both to happen at the same time. Another name for disjoint events is mutually exclusive. This section is relatively straightforward, so these notes will be rather short.

In the following discussion, the capital letters E and F represent possible outcomes from an experiment, and P(E) represents the probability of seeing outcome E.

For disjoint events, the outcomes of E or F can be listed as the outcomes of E followed by the outcomes of F. The Addition Rule for the probability of disjoint events is:

(E or F)=(E) + (F)

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

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

The addition rule only applies to events that are disjoint. If two (or more) events are not disjoint, then this rule must be modified because some outcomes may be counted more than once. For the formula (E or F) = (E) + (F), all the outcomes that are in both E and F will be counted twice. Thus, to compute P (E or F), these double-counted outcomes must be subtracted (once), so that each outcome is only counted once.

The General Addition Rule is:

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

where P (E and F) is the set of outcomes in both E and F. This rule is true both for disjoint events and for non-disjoint events, for if two events are indeed disjoint, then P (E and F) = 0, and the General Addition Formula simply reduces to the basic addition formula for disjoint events.

Example

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

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

E and F are not disjoint because there is one card that is both a queen AND a heart, so we must use the General Addition Rule. We know the following probabilities using the classical (counting, equally-likely outcomes) method:

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

Therefore,

Equation
Finally, it is often easier to calculate the probability that something will not happen rather than determining the probability that it will happen. The complement of the event E is the “opposite” of E. We write the complement of outcome E as Ec. The complement E^c consists of all the outcomes that are not in that event E

For example, when rolling one die, if event = {even number}, then E= {odd number}. If  event = {1,2}, then Fc = {3, 4, 5, 6}.

It should make sense that the probability of the complement Ec occurring is just 1 minus the probability that event E occurs. In formula form:

P(Ec ) = 1 – P(E)