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Section 2.4
Introduction to Probability
S = sample space – the set of all possible outcomes for a given experiment
(Note: This is different than s, which represents the standard deviation of a sample. There
IS a difference between using an upper-case S and a lower-case s.)
Examples: Suppose you roll a die.
1) Let S represent the possible numbers you can roll.
Therefore, S = {1, 2, 3, 4, 5, 6}
2) Let S represent whether or not the number is even or odd.
Therefore, S = {even, odd}
Examples: Suppose you flip a coin twice.
3) Let S represent the possible “faces” you can get.
Therefore, S = {HH, HT, TH, TT} where H = Head and T = Tail
4) Let S represent the number of tails you can get
Therefore, S = {0, 1, 2}
Event – any subset of the sample space. Events are generally denoted by capital letters near the
beginning of the alphabet. However, any capital letter (or combination of capital letters) is
appropriate, with the exception of S (since it is reserved for the sample space.)
Examples: Suppose we return to example 1.
5) Let A be the event you roll a power of 2. Then A = {2, 4}.
6) Let B be the event you roll a prime number. Then B = {2, 3, 5}.
Complement of an Event: A = the set of all outcomes in S that are not in A.
Other complement notation: A  AC  A
For example 5, A = {1, 3, 5, 6} and for example 6, B = {1, 4, 6}
Union of two Events: A  B = the set of all outcomes in S that are in A or in B.
Note: Unless otherwise specified, any “or” is assumed to be an inclusive “or”.
Inclusive means that the phrase “in A or in B” actually means “in A or in B or in both”
Exclusive would mean “in A or in B” meant “in A or in B but not in both”
Using A and B from examples 5 and 6, A  B = {2, 3, 4, 5}
Intersection of two Events: A  B = the set of all outcomes in S that are in A and in B.
Using A and B from examples 5 and 6, A  B = {2}
Empty Set:  Use this notation if a given event does not contain any of the outcomes in S. For
example, using example 1, suppose C = the event of getting a number greater than 20.
Then C =  .
Note: C =  is NOT the same as C = {  }.
C =  contains no outcomes whereas C = {  } contains ONE outcome.
Disjoint Events: Events A and B are said to be disjoint events if A  B = 
Mutually Exclusive Events:
If you are dealing only with two events, “mutually exclusive” and “disjoint” are
synonymous.
If you are dealing with more than two events, then “mutually exclusive” is stronger than
“disjoint”. “Disjoint” is only used to describe the relationship between two sets.
Take the three events A, B, and C.
If we say that A, B, and C are mutually exclusive events, then the following are implied:
A  B =  , A  C =  , and B  C =  .
If only A  B =  and A  C =  are true, then we would say that A and B are disjoint and
A and C are disjoint. Note we are no longer claiming that B  C =  .
Venn Diagram: A “picture” or diagram representation of events.
DeMorgan’s Laws:
A B  A  B
and
A B  A  B
n
 n

Extended DeMorgan’s Laws:   Ai    Ai
 i 1  i 1
and
 n  n
  Ai    Ai


 i 1  i 1
Probability of Event A Notation: P(A)
Probability Rules/Properties:
1. P(S) = 1.
2. P(  ) = 0.
3. If A is an event, then the probability of A must satisfy 0 ≤ P(A) ≤ 1.
4. For any event A, P( A ) = 1 – P(A).
5. If A and B are disjoint events, then P(A  B) = 0.
6. If A  B, then P(A) ≤ P(B).
7. If A and B are disjoint events, then P(A  B) = P(A) + P(B).
8. In general, P(A  B) = P(A) + P(B) – P(A  B).
9. In general, P(A  B  C) = ??? (Try to finish this equation before class on Thursday.)
10. For any two events, A and B, P(B) = P(B ∩ A) + P(B ∩ A ).
(This property is known as the Law of Total Probability.)
Something to keep in mind with respect to notation…
A, A ,  , S, A  B, A  B, etc. are sets, not numbers.
But…
P(A), P( A ), P(  ), P(S), P(A  B), P(A  B) are probabilities and therefore are numbers.
Additional Examples:
Ex 7: A box contains seven marbles: two green, three blue, one red, and one white. If one
marble is drawn at random, what is the probability that the marble drawn is a blue or a
red marble?
Ex 8: A real estate agent is showing homes to a prospective buyer. There are ten homes in the
desired price range listed in the area. The buyer has time to visit only three of them.
a) In how many ways could the three homes be chosen if the order of visiting is
considered?
b) In how many ways could the three homes be chosen if the order of visiting is
disregarded?
c) If four of the homes are new and six have previously been occupied and if the three
homes to visit are randomly chosen, what is the probability that all three are new?
(The same answer results regardless of whether order is considered. This is another
reason why we generally do not care about order.)
Ex 9: Suppose we have a group of 50 animals, 10 males and 40 females. We select a group of
5 animals at random. What is the probability that exactly 3 females are chosen? At least
3 females are chosen?
Ex 10: Use the Venn Diagram below to answer the following questions.
(The Venn Diagram shows the desserts people ordered at a party.)
Ice
Cream
7
15
21
Cake
9
a) What is the probability that a person had both ice cream and cake for dessert?
b) What is the probability that a person had cake for dessert?
Ex 11: Suppose that P(A) = .7, P(B) = .5, and P(A  B) = .2. Draw a Venn Diagram that
represents this information.
Ex 12: Refer to the following Venn Diagram for events A and B. Find each of the following
probabilities.
A
.5
B
.3
.1
.1
.1
a) P(A  B)
b) P(A  B)
c) P(A  B )
d) P(A  B )
e) P( A  B )
f) P(A)
Ex 13: For any two sets, how many disjoint subsets would you need to form the entire sample
space S? What are the disjoint subsets?
(Hint: Use an extension of the Law of Total Probability.)
Ex 14: For any three sets, how many disjoint subsets would you need to form the entire
sample space S?
Ex 15: For any n sets, how many disjoint subsets would you need to form the entire sample
space S?