• Warm up: Distance Learning
Age group (yr):
Probability:
18 to 23
24 to 29
30 to 39
40 or over
0.57
0.17
0.14
0.12
(a)Show that this is a legitimate probability model.
Each probability is between 0 and 1 and
0.57 + 0.17 + 0.14 + 0.12 = 1
(b)Find the probability that the chosen student is not
in the traditional college age group (18 to 23
years).
P(not 18 to 23 years) = 1 – P(18 to 23
years)
= 1 – 0.57 = 0.43
Probability Rules
Distance-learning courses are rapidly gaining popularity among
college students. Randomly select an undergraduate student who
is taking distance-learning courses for credit and record the
student’s age. Here is the probability model:
Suppose a six-sided die is rolled. The
event that the die would land on an even
number would be E = {2, 4, 6}
The event that the die would land on a
prime number would be P = {2, 3, 5}
What would be the event E or P
happening?
E or P = {2, 3, 4, 5, 6}
This is an example of the union of two events.
Let’s revisit rolling a die and getting an
even or a prime number . . .
E or P = {2, 3, 4, 5, 6}
E or P would be any number in either
Another waycircle.
to represent this
Diagram.
Even number
Why is the number
1 outside the
circles?
4
is with a Venn
Prime number
3
2
6
1
5
General Rule for Addition
Since the intersection is added in twice, we subtract out the
intersection.
For any two events A and B,
P( A  B)  P( A)  P( B)  P( A  B)
A
B
Example:
Probability of a Promotion
Debra and Matt are waiting word on whether they have been
made partners in their law firm.
Debra guesses their probabilities as:
P(D) = 0.7
P(M) = 0.5
P(both promoted) = 0.3
P(at least one is promoted)
= P(D or M)
= P(D)
+ P(M) – P(D and M)
= 0.7 + 0.5 - 0.3
= 0.9
What is the probability neither is promoted?
This is is the complement of
P(at least one is promoted)c
= 1 – 0.9
= 0.1
Venn diagram and
probabilities
Discuss joint probabilities in diagram: P(D and
MC)
Example cont.
(working with joint events)
• Construct a table and write in the probabilities
Debra assumes.
• Fill in rest of table.
Matt
Promoted
Debra
Promoted
Not Prom
Total
0.3
0.2
0.5
Not Prom
0.4
0.1
0.5
Total
0.7
0.3
1.0
Probability Rules
Venn Diagrams and Probability
Because Venn diagrams have uses in other branches of
mathematics, some standard vocabulary and notation
have been developed.
The complement AC contains exactly the outcomes that are not in A.
The events A and B are mutually exclusive (disjoint) because
they do not overlap. That is, they have no outcomes in common.
The intersection of events A and B (A ∩ B) is the set of all
outcomes in both events A and B.
Probability Rules
• Venn Diagrams and Probability
The union of events A and B (A ∪ B) is the set of all outcomes in
either event A or B.
Exercise
1. Customers at a restaurant are offered a choice of chips
or jacket potato to go with either lasagna or pizza.
Out of a group of 16, 11 have the lasagna. 7 choose
chips to accompany their meal. 5 of those who choose
chips have the lasagna. What is the probability that
one chosen at random has neither chips nor lasagna?
Solution:
L: lasagna
C: chips
L
C
Exercise
1. Customers at a restaurant are offered a choice of chips
or jacket potato to go with either lasagna or pizza.
Out of a group of 16, 11 have the lasagna. 7 choose
chips to accompany their meal. 5 of those who choose
chips have the lasagna. What is the probability that
one chosen at random has neither chips nor lasagna?
Solution:
L: lasagna
16
C: chips
P(no chips, no lasagna)
3

16
C
L
6
5
2
3
e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are
yellow and 3 are both. What is the probability if a plant is
picked at random that it is (a) either a petunia or yellow
(b) either a petunia or yellow but not both?
Solution:
Let P be event ”Petunia” and Y be event “Yellow”
(a) We want to find P(P or Y)
P
Y
e.g.2 In a group of 15 mixed plants, 6 are petunias, 8 are
yellow and 3 are both. What is the probability if a plant is
picked at random that it is (a) either a petunia or yellow
(b) either a petunia or yellow but not both?
Solution:
Let P be event ”Petunia” and Y be event “Yellow”
(a) We want to find P(P or Y)
15
Y
P
3
3
5
4
11
P(P or Y) =
15
(b) 8
15
Example 2: Venn diagram for 3 events
Question
At Dawnview High there are 400 Grade 11 learners. 270 do
Computer Science, 300 do English and 50 do Business studies. All
those doing Computer Science do English, 20 take Computer
Science and Business studies and 35 take English and Business
studies. Using a Venn diagram, calculate the probability that a
pupil drawn at random will take:
•
•
•
•
English, but not Business studies or Computer Science
English but not Business studies
English or Business studies but not Computer Science
English or Business studies
•
•
•
•
English, but not Business studies or Computer Science
English but not Business studies
English or Business studies but not Computer Science
English or Business studies
•
•
•
•
English, but not Business studies or Computer Science
English but not Business studies
English or Business studies but not Computer Science
English or Business studies
• English, but not Business studies or Computer
Science
•
The count in this region is 15 and there are a total of 400 learners in the grade. Therefore the
probability that a learner will take English but not Business studies or Computer Science is
15400=380.
• English but not Business studies
•
The count in this region is 265. Therefore the probability that a learner will take English but not
Business studies is 265400=5380.
• English or Business studies but not Computer Science