3.5. Additive and Multiplicative Rules http://www.ck12.org
Note thatP(E) =P(C∪D). Remember, eventCconsists of 13 cards and eventDconsists of 12 face cards. Event
P(C∩D)consists of the 3 face-spade cards: The king, jack and, queen of spades cards. Using the additive rule of
probability formula,
P(A∪B) =P(A)+P(B)−P(A∩B)
=
13
52
+
12
52
−
3
52
= 0. 250 +. 231 −. 058
= 0. 423
= 42 .3%
I hope that you have learned, through this example, the reason why we subtractP(C∩D). It is because we do not
want to count the face-spade cards twice.
Example:
If you know that 84.2% of the people arrested in the mid 1990’s were males, 18.3% are under the age of 18, and
14 .1% were males under 18. What is the probability, that a person selected at random from all those arrested, is
either male or under 18?
Solution:
Let
A={person selected is male}
B={person selected is under 18}From the percents given,
P(A) = 0. 842 P(B) = 0. 183 P(A∩B) = 0. 141
The probability of a person selected is male or under 18P(A∪B):
P(A∪B) =P(A)+P(B)−P(A∩B)
= 0. 842 + 0. 183 − 0. 141
= 0. 884
= 88 .4%
This means that 88.4% of the people arrested in the mid 1990’s are either males or under 18.
It happens sometimes thatA∩Bcontains no simple events, i.e.,A∩B={φ}, the empty set. In this case, we say that
the eventsAandBaremutually exclusive.
Definition
IfA∩Bcontains no simple events, thenAandBaremutually exclusive.The figure below is the Venn diagram of mutually exclusive events, for example setAmight represent all the
outcomes of drawing a card, and setBmight represent all the outcomes of tossing three coins.