Download M118 SECTION 8.2 - UNION, INTERSECTION, and COMPLEMENT

M118 SECTION 8.2 - UNION, INTERSECTION, and COMPLEMENT ; ODDS
1)
Remember from Section 7.2:
A and B are two events in sample space S then:
UNION: A ∪ B = x∣ x∈A or x∈B
INTERSECTION: A ∩ B = x∣ x∈A and x∈B
We also define event A or B to be A∪ B and the event A and B to be A∩B
Ex: Experiment- Roll a fair die
S = 1, 2, 3, 4, 5, 6
a) What is the probability of rolling an odd number and exactly
divisible by 3
Let A = event of rolling an odd number
B= exactly divisible by 3
so P(roll an odd number and exactly divisible by 3) = A ∩ B = 3
n( A ∩ B) 1
P(A ∩ B) =
=
n(S)
6
b) probability of odd or exactly divisible by 3
(A ∪ B) = 1, 3, 5, 6
n(A ∪ B) 4
2
P ( A ∪ B) =
=
=
n(S)
6
3
c) probability of odd and prime
A= event odd = 1, 3, 5
2 1
P(A ∩ C) = =
6 3
C = event prime = 2, 3, 5
d) probability of odd or prime
4 2
P(A ∪ C) = =
6 3
1
THEOREM Probability of a Union of Two Events
For any events A and B,
P(A∪B) = P(A) + P(B) - P(A∩B)
If A and B are disjoint, then
P(A∪B) = P(A) + P(B)
Ex: Suppose 2 fair dice are rolled:
a) WHat is the probability of a sum of 7 or 11?
b) What is the probability that both dice are same or a sum less than 5?
c) What is the probability of a sum of 2 or 3 turns up?
d) What is the probability that both numbers are the same or sum greater than 8?
2
What is the probability that a number selected at random from the first 500 positive integers
is exactly divisible by 4 or 6?
COMPLEMENT
Suppose we divide a sample space S is divided into 2 subsets E and E' such that
E ∩ E' = ∅ and E ∪ E' = S
Then E is called the complement of E' and
P(S) = P(E ∪ E') = P(E) + P(E') = 1
so P(E) = 1 - P(E')
OR
P(E') = 1 - P(E)
If the probability of rain is 65% then the probability of no rain is 35% (1 - .65).
Sometimes it is easier to find the probability of the complement then the probability of the
event.
P(at least 1 thing) = P(one or more things) = 1 - P(0 things)
Prob of at least 1 man on a committee = 1 - the probability of 0 men on the committee
P(1 man) = 1 - P(0 men)
3
Ex: A shipment of 40 precision parts, including 8 that are defective, is sent to an assembly
plant. The quality control division selects 10 at random for testing and rejects the whole
shipment if 1 or more in the sample are found defective. What is the probability that the
shipment will be rejected?
Example:
A poll was conducted preceding an election to determine the relationship between voter
persuasion concerning a controversial issue and the area of the city in which the voter
lives. Five hundred registered voters were interviewed from three areas of the city. The data
are shown below.
Area of City
East
North
Inner
Voter Opinion
Favor
Oppose
30
45
25
65
125
70
No opinion
35
40
65
Compute the probability of having no opinion on the issue or living in the inner city.
Compute the probability of a voter in favor of the issue and living in the north region of the
city.
4
ODDS: In gaming situations, if we know the probability of something then we may want to
know the odds for or against.
PROBABILITY → ODDS
If P(E) is the prob of E
Odds for E is
P(E)
P(E)
n(E)
=
=
1 - P(E) P(E') n(E')
Odds against E is
P(E')
n(E')
=
P(E)
n(E )
Odds are expressed as a ratio of whole numbers.
Ex: 5:3
or
5
3
or
5 to3
NEVER express a probability with a colon (:)
What are the odds for rolling a sum of 8 in a single roll of 2 dice?
ODDS → PROBABILITY
If odds for E are a/b then P(E) =
a
a+b
a) The odds for rolling a 5 before a 7 are 2/3, what is the probability of rolling a 5 before
a 7?
b) The odds against rolling a 6 before a 7 are
before a 7?
5
6
. What is the probability of rolling a 6
5
2) In
a certain town, 4% of people commute to work by bicycle. If a person is selected
randomly from the town, what are the odds against selecting someone who commutes by
bicycle?
3)
Fill in the blanks:
PROBABILITY
ODDS FOR
ODDS AGAINST
3/8
1/4
.4
.55
4/7
11/9
4:1
49 to 51
3/5
11/7
4/3
8:9
6