Probability Rules

• Warm up: Distance LearningDistance-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:

Age group (yr): 18 to 23 24 to 29 30 to 39 40 or over

Probability: 0.57 0.17 0.14 0.12

(a)Show that this is a legitimate probability model.

(b)Find the probability that the chosen student is not in the traditional college age group (18 to 23 years).

Each probability is between 0 and 1 and 0.57 + 0.17 + 0.14 + 0.12 = 1

P(not 18 to 23 years) = 1 – P(18 to 23 years)

= 1 – 0.57 = 0.43

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}

Another way to represent this is with a Venn Diagram.

Even number

24

6

Prime number

3

5

1

E or P would be any number in either circle.

Why is the number 1 outside the

circles?

General Rule for Addition

For any two events A and B,

( ) ( ) ( ) ( )P A B P A P B P A B

A B

Since the intersection is added in twice, we subtract out the intersection.

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) = 0.7P(M) = 0.5P(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 ofP(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 Not Prom Total

Debra Promoted 0.3 0.7

Not Prom

Total 0.5 1.0

0.4

0.2 0.1

0.5

0.3

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.

Probability Rules

• Venn Diagrams and Probability

The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B.

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?

CLSolutio

n:L: lasagna

C: chips

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?

16 CL

5 26

3

Solution:L:

lasagna

C: chips

P(no chips, no

lasagna) 16

3

Exercise

Solution: Let P be event ”Petunia” and Y be event

“Yellow”

YP

(a) We want to find P(P or 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?

15

Solution: Let P be event ”Petunia” and Y be event

“Yellow”

YP

3 53

4

P(P or Y) =15

11

(a) We want to find P(P or 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?

(b) 15

8

Example 2: Venn diagram for 3 eventsQuestionAt 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