78 5. Combinatorial Probability
the subsetL={ 4 , 5 , 6 }⊆Scorresponds to the event that we throw a
number larger than 3.
The intersection of two subsets corresponds to the event that both events
occur; for example, the subsetL∩E={ 4 , 6 }corresponds to the event that
we throw a better-than-average number that is also even. Two eventsA
andB(i.e., two subsets ofS) are calledexclusiveif they never occur at
the same time, i.e.,A∩B=∅. For example, the eventO={ 1 , 3 , 5 }that
the outcome of tossing a die is odd and the eventEthat it is even are
exclusive, sinceE∩O=∅.
5.1.1What event does the union of two subsets corresponds to?So letS={s 1 ,s 2 ,...,sk}be the set of possible outcomes of an experi-
ment. To get a probability space we assume that each outcomesi∈Shas
a “probability”P(si) such that
(a)P(si)≥0 for allsi∈S,and
(b)P(s 1 )+P(s 2 )+···+P(sk)=1.Then we callS, together with these probabilities, aprobability space.For
example, if we toss a “fair” coin, thenP(H)=P(T)=^12. If the dice in our
example is of good quality, then we will haveP(i)=^16 for every outcomei.
A probability space in which every outcome has the same probability
is called auniform probability space. We shall only discuss uniform spaces
here, since they are the easiest to imagine and they are the best for the
illustration of combinatorial methods. But you should be warned that in
more complicated modeling, nonuniform probability spaces are very often
needed. For example, if we are observing whether a day is rainy or not, we
will have a 2-element sample spaceS={RAINY,NONRAINY}, but these
two will typicallynothave the same probability.
The probability of an eventA⊆Sis defined as the sum of probabilities of
outcomes inA, and is denoted byP(A). If the probability space is uniform,
then the probability ofAis
P(A)=|A|
|S|
=
|A|
k.
5.1.2Prove that the probability of any event is at most 1.5.1.3What is the probability of the eventEthat we throw an even number
with the die? What is the probability of the eventT ={ 3 , 6 }that we toss a
number that is divisible by 3?
5.1.4Prove that ifAandBare exclusive, thenP(A)+P(B)=P(A∪B).5.1.5Prove that for any two eventsAandB,
P(A∩B)+P(A∪B)=P(A)+P(B).