COUNTING METHODS AND PROBABILITY

1
MGF1106
Unit Four Ex. 6.6 Probability Of Independent Or
Dependent Events Objectives 19-21
2
REVIEW
If a student is selected at random from these
100, what is the probability that the student
made an A?
100
3
Exercise 6.6
Objective 19
To find the probability of independent events
4
independent
Two events, A and B, are _____________ if the
occurrence or non-occurrence of one does not
change the probability that the other will occur.
If two events are independent, then P(A and B)
_________.
P(A)P(B)
5
Two jaw breakers are selected from a bag of jaw
breakers containing 4 green, 5 red and 8 yellow
jaw breakers. The color of the jaw breaker is
noted and the jaw breaker is returned to the bag.
A second jaw breaker is then selected from the
bag. Find the probability that both jaw breakers
are red.
The events are independent since the first jaw
breaker is replaced before the second one is
selected.
6
4 G 5 R 8 Y
Let A be the event that the first is red.
Let B be the event that the second is red.
17 T
5
P(A)
17
5
P(B)
17
P(A and B) P(A)P(B).
7
4 G 5 R 8 Y
Find the probability that the first is yellow and
the second is green
17 T
Let A be the event that the first is yellow.
Let B be the event that the second is green.
8
4
P(A)
P(B)
17
17
P(A and B) P(A)P(B).
8
4 G 5 R 8 Y
Find the probability that both are of the same
color.
2G or 2 R or 2Y
17 T
P(2G)
P(2R)
P(2Y)
P(2G or 2R or 2Y)
9
The probability that Sam will pass the CLAST is
0.25 and the probability that Betty will pass the
CLAST is 0.60.
Find the probability that Sam will pass and Betty
will not.
p
p
Sam passes is 0.25.
p
f
Betty does not is 0.40.
p
f
f
f
Sam passes and Betty does not
(0.25)(0.40)
.1000
10
Find the probability that at least one of them
will pass.
p
p
p
f
p
f
f
f
(0.25)(0.60)
(0.25)(0.40)
(0.75)(0.60)
0.15 .10 .45
0.70
11
Exercise 6.6
Objective 20
To find the probability of dependent events
12
If the probability of event occurring is affected
by the probability of another event having
occurred, then the events are said to be
__________.
dependent
13
The probability that GCCC basketball team will
win the regional competition is 0.80. If the
team wins the regional competition, then the
probability that the team will win the state
competition is .60. Find the probability that
the team will win the regional and state
competition.
P(A) .80
Let A event GCCC wins the regional
P(B,given A) .60
Let B event GCCC wins the state
14
A jar contains 5 red and 6 blue marbles. Three
marbles are selected at random without
replacement. Find the probability of selecting 3
blue marbles.
5R 6B
Without replacement means that each draw is
dependent on the results of the previous draw.
15
Method I
A jar contains 5 red and 6 blue marbles. Three
marbles are selected at random without
replacement. Find the probability of selecting 3
blue marbles.
5R 6B
P(1st marble is blue)
P(2nd marble is blue, given 1st is blue)
P(3rd marble is blue, given 1st two are blue)
P(all 3 are blue)
16
Method II
A jar contains 5 red and 6 blue marbles. Three
marbles are selected at random without
replacement. Find the probability of selecting 3
blue marbles.
Number of ways of getting 3 blue marbles.
6C3
20
Total number of ways of getting 3 marbles.
11C3
165
P(3 B)
17
METHOD I
Two people are to be selected at random from a
group of four Math professors and five English
professors to attend a convention in Orlando.
Find the probability that two Math professors are
chosen.
P(1ST person is a math professor)
P(2nd person is a math professor, given the 1st
person is a math professor)
18
Method II
Two people are to be selected at random from a
group of four Math professors and five English
professors to attend a convention in Orlando.
Find the probability that two Math professors are
chosen.
How can 2 Math professors be chosen from 4 Math
professors?
4C2
19
How many ways are there of choosing 2 people form
this group?
9C2
4C2
6
9C2
36
20
Method I
Two people are to be selected at random from a
group of four math professors and five English
professors to attend a convention in Orlando.
Find the probability that one English and one
math professor will be chosen.
21
Method II
Two people are to be selected at random from a
group of four Math professors and five English
professors to attend a convention in Orlando.
Find the probability that two Math professors are
chosen.
How can 2 Math professors be chosen from 4 Math
professors?
4C2
22
Method II
Two people are to be selected at random from a
group of four Math professors and five English
professors to attend a convention in Orlando.
Find the probability that one English and one
math professor will be chosen.
1 E and 1 M
5C1 4C1
Total number of ways of selecting 2 people from 9
9C2
23
5C1 4C1
9C2
20
36
24
A recent survey indicated that ten percent of
students who enter the community college receive
an AA degree. Of these, five percent enter a
major university. What is the probability that a
randomly selected community college student will
receive an AA degree and enter a major university?
E receive an AA degree
M enter a major university given he received
an AA degree
P(E and M)
0.10(0.05)
0.0050
25
Exercise 6.6
Objective 21
To find the probability of an event, given that
another event has occurred
26
Solve for P(B,given A) in the formula
27
The following table shows the grade distribution
of a course by sex and and grade. The course had
a total of 100 students.
28
What is the probability that the student made a
B, given the student is a female?
Using the formula
29
What is the probability that the student made a
B, given the student is a female?
Using a reduced sample space (females)
5
30
What is the probability that the student is a
male, given the student made an F?
5
31
The probability that Sarah drinks coffee is 0.45
and the probability that she drinks coffee and
has dessert is 0.20. Find the probability that
she has dessert, given that she drinks coffee.
C
D
.20
.25