Sum Rule
Assume that there are two events, A and B. The sum rule determines either the
probability that event A or event B occurs, or the probability that both occur.
There are two possibilities –
that the two events are mutually exclusive
(disjoint) or not mutually exclusive (conjoint).
Mutually Exclusive
If events A and B are mutually exclusive then:
P(A or B) = P(A) + P(B).
As an illustration, assume that there is
a group of 150
students where 30 are
freshmen and 60
are sophomores. Find the probability that a student picked
from this group at random is either a freshman or sophomore. In this case the
individual probabilities are:
P(freshman) = 30/150
P(sophomore) = 60/150
Therefore, P(freshman or sophomore) = 30/150 + 60/150 = 90/150. In other
words 90 of the 150 students are freshmen or sophomores.
Not Mutually Exclusive
If events A and B are not mutually exclusive then
P(A or B) = P(A) + P(B) – P(A and B)
As an illustration, assume that there is
a group of 150
students where 40 are
juniors, 50 are female, and 30 are both female and juniors. The task is to find
the probability that a student picked from this group
at random is either a
junior or female.
Here the individual probabilities are as follows:
P(junior) = 40/150
P(female) = 50/150
P(junior and female) = 30/150
Thus:
P(junior or female) = 40/150 + 50/150 – 30/150 = 60/150
This makes sense since 90 of the 150 students are juniors or female. The point
to subtracting the percentage of students who are junior and female is to avoid
double counting. When we add 40 juniors to 50 females and get a total of 90,
we have over-counted. The 30 female juniors were counted twice; 90 minus 30
makes 60 students who are juniors or female.
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