Exercise: Birth and death rates !  The estimated beginning 2006 values for China:

"  Population: 1,313,973,713 "  Birth rate: 13.25 per 1,000 "  Death rate: 6.97 per 1,000

(a)  Approximately how many births were there in China in 2006? (b)  About how many deaths were there in China in 2006? (c)  Based on births and deaths alone, about how much did the

population of China rise during 2006? (d)  Ignoring immigration and emigration, what is the 2006 rate of

population growth of China? What is the population growth rate expressed as a percentage?

1

Exercise: Birth and death rates (a)  Approximately how many births were there in China in 2006?

(b)  About how many deaths were there in China in 2006?

2

Exercise: Birth and death rates (c) Based on births and deaths alone, about how much did the population of China rise during 2006? (d)Ignoring immigration and emigration, what is the 2006 rate of population growth of China? What is the population growth rate expressed as a percentage?

3

4

6.5 Combining Probabilities (part 1)

! What is the probability that event A and event B both occur.

!  We call the probability of event A and event B occurring a joint probability.

! What is the probability that event A occurs given that I know event B has occurred?

!  This relates to a conditional probability.

! What is the probability that either event A or event B (or both) occurs?

Example 1: Roll two 6-sided dice !  Let A be the event of rolling a sum of 7. !  Let B be the event of rolling a doubles.

" What is the probability that event A occurs?

" What is the probability that event B occurs?

" What is the probability that event A or B (or both) occurs?

5

Answers: " What is the probability

that event A occurs? 6/36=1/6

" What is the probability that event B occurs? 6/36=1/6

" What is the probability that event A or B (or both) occurs?

12/36=1/3 6

Let A be the event of rolling a sum of 7. Let B be the event of rolling a doubles.

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Example 1: Roll two 6-sided dice !  Let A be the event of rolling a sum of 7. !  Let B be the event of rolling a doubles.

" For this example, events A and B are

" When two events can not possibly occur at the same time, we say they are Non-overlapping events.

" Non-overlapping Events are also called Mutually Exclusive events.

7

Non-overlapping events

8

Non-overlapping Events ! Example 2: Roll two 6-sided dice

" Event A: You roll two even numbers " Event B: At least one of the numbers is a 5

Ways A could happen: {(2,2),(2,4),(2,6),(4,2),(4,4), (4,6),(6,2),(6,4),(6,6)}

Ways B could happen: {(1,5),(2,5),(3,5),(4,5),(5,5),(6,5), (5,1),(5,2),(5,3),(5,4),(5,6)}

Nothing in common

Events A and B are non-overlapping events here.

These can NOT both happen on

a single roll.

9

Non-overlapping Events ! As a Venn diagram, we show

non-overlapping events A and B as…

A B

Events A and B have nothing in

common. They do not

overlap, so the circles do not

intersect.

10

Non-overlapping Events ! Complements are non-overlapping

events (i.e. complements are mutually exclusive events).

! Example 3: Roll a die " Event A: You roll an even " Event B: You roll an odd

! There is no roll on which both A and B can both happen.

B

A 2 4 6

1 2 3

11

Overlapping Events ! Two events are overlapping if they CAN

occur together.

! Example 4: Roll two 6-sided dice " Event A: You roll two even numbers " Event B: At least one of the numbers is a 4

Here, A and B are overlapping events. They CAN both occur. There is at least one outcome that qualifies for both events.

These CAN both happen on

a single roll.

12

Overlapping Events ! Example 4: Roll two 6-sided dice

" Event A: You roll two even numbers " Event B: At least one of the numbers is a 4

Ways A could happen: {(2,2),(2,4),(2,6),(4,2),(4,4), (4,6),(6,2),(6,4),(6,6)}

Ways B could happen: {(1,4),(2,4),(3,4),(4,4),(5,4),(6,4), (4,1),(4,2),(4,3),(4,5),(4,6)}

Some in common

The bolded blue outcomes are ones in which BOTH events have occurred.

! Example 4: Roll two 6-sided dice " Event A: You roll two even numbers " Event B: At least one of the numbers is a 4 " P(both A and B occur) = 5/36

Overlapping Events

13

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Roll number 1

Roll number 2

Five of the 36 possible outcomes qualify under both event A and event B.

! As a Venn diagram, we show overlapping events A and B as…

Overlapping Events

14

A B

At least some in common

In A but not in B In B but

not in A

Not in A or B

Overlapping Events ! Example 4: Roll two 6-sided dice (36 possible outcomes)

" Event A: You roll two even numbers " Event B: At least one of the numbers is a 4

15 15

A B

(2,4),(4,2),(4,4),(4,6),(6,4) (2,2),(2,6),

(6,2),(6,6) (1,4),(3,4),(5,4),(4,1),(4,3),(4,5)

(1,1),(1,2),(1,3),(1,5),(1,6), (2,1),(2,3), (2,5),

(3,1), (3,2),(3,3), (3,5),(3,6), (5,1),(5,2),

(5,3)(5,5),(5,6),(6,1),(6,3),(6,5)

Why make such a big deal in defining Non-overlapping and Overlapping Events? Because the formula for how we calculate Either/Or probabilities is different for these two things.

16

Either/or Probabilities for Non-overlapping and Overlapping Events

17

A B

A B

Without the subtraction, the

overlap would be counted twice

!  Let A and B be non-overlapping events, then P(A or B) = P(A) + P(B)

!  Let A and B be overlapping events, then

P(A or B) = P(A) + P(B) – P(A and B)

Either/or Probabilities for Non-overlapping Events ! Example 3: Roll a die

" Event A: You roll an even " Event B: You roll an odd

" A and B are non-overlapping events, so P(A or B)=P(A) + P(B) = 0.5 + 0.5 = 1.0

18

(Clearly we’re 100% sure at least one of

them happens.)

These can NOT both happen on

a single roll.

Either/or Probabilities for Non-overlapping Events ! Example 2: Roll two 6-sided dice

" Event A: You roll two even numbers " Event B: At least one of the numbers is a 5

" A and B are non-overlapping events, so P(A or B) = P(A) + P(B)

= 9/36 + 11/36 = 20/36 = 5/9 19

9 ways

11 ways

! Example 2: Roll two 6-sided dice " Event A: You roll two even numbers " Event B: At least one of the numbers is a 5 " P(A or B) = 20/36 = 5/9

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Either/or Probabilities for Non-overlapping Events

20

Roll number 1

Roll number 2

These can NOT both happen on

a single roll.

Either/or Probabilities for Overlapping Events

21

9 ways

11 ways But there’s overlap!!!!

How many rolls qualify under BOTH events? 5

! Example 4: Roll two 6-sided dice " Event A: You roll two even numbers " Event B: At least one of the numbers is a 4

" P(A or B) = P(A) + P(B) - P(A and B) = 9/36 + 11/36 – 5/36 = 15/36 = 5/12

! Example 4: Roll two 6-sided dice " Event A: You roll two even numbers " Event B: At least one of the numbers is a 4 " P(A or B) = 9/36 + 11/36 – 5/36 = 15/36 = 5/12

Either/or Probabilities for Overlapping Events

22

Roll number 1

Roll number 2

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

These can both happen

on a single roll.

Either/or Probabilities ! Example 5: To improve tourism between France and the United States, the two governments form a committee consisting of 20 people: 2 American men, 4 French men, 6 American women, and 8 French women (as shown in Table 6.9). If you meet one of these people at random, what is the probability that the person will be either a woman or a French person? 23

Twelve of the 20 people are French, so the probability of meeting a French person is 12/20. Similarly, 14 of the 20 people are women, so the probability of meeting a woman is 14/20. However the eight French women were included in both groups. P(woman or French) = + – = = 12

20 8 20

14 20

18 20

probability of a woman

probability of a French person

probability of a French woman

9 10

24

P(A or B) = P(A) + P(B) - P(A and B)

25

Summary of combining probabilities for Either/or Probabilities

Either/or Probabilities for Non-overlapping Events !  Let A and B and C all be non-overlapping

events, then P(A or B or C)=P(A) + P(B) + P(C)

26

A B

C

This can be extended to any number of events provided they are ALL non-overlapping.

Either/or Probabilities for Overlapping Events

27

A B

C

!  Let A and B and C all be overlapping events, then

P(A or B or C)=P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C)