Section 7-2Addition Counting Principles

Warm-upHow many positive integers less than or equal to 1000

satisfy each condition?

a. Divisible by 5? b. Divisible by 7?

Divisible by 5 or 7?

Warm-upHow many positive integers less than or equal to 1000

satisfy each condition?

a. Divisible by 5? b. Divisible by 7?

Divisible by 5 or 7?

200

Warm-upHow many positive integers less than or equal to 1000

satisfy each condition?

a. Divisible by 5? b. Divisible by 7?

Divisible by 5 or 7?

200 142

Warm-upHow many positive integers less than or equal to 1000

satisfy each condition?

a. Divisible by 5? b. Divisible by 7?

Divisible by 5 or 7?

200 142

314

Union:

Union: Values that are in one set or another; does not need to be part of both

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Disjoint/Mutually Exclusive:

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Disjoint/Mutually Exclusive: A situation where two or more sets have nothing in common

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Disjoint/Mutually Exclusive: A situation where two or more sets have nothing in common

i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t have them both happen at the same time

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Disjoint/Mutually Exclusive: A situation where two or more sets have nothing in common

i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t have them both happen at the same time

Intersection:

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Disjoint/Mutually Exclusive: A situation where two or more sets have nothing in common

i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t have them both happen at the same time

Intersection: Values that are shared by two or more sets

Union: Values that are in one set or another; does not need to be part of both

Notation: A B

Disjoint/Mutually Exclusive: A situation where two or more sets have nothing in common

i.e. Rolling a sum of 2 or 3 on a pair of dice; you can’t have them both happen at the same time

Intersection: Values that are shared by two or more sets

Notation: A B

Addition Counting Principle (Mutually Exclusive Form):

Addition Counting Principle (Mutually Exclusive Form):

If two finite sets A and B are mutually exclusive, then

N(A B) = N(A)+ N(B)

Addition Counting Principle (Mutually Exclusive Form):

If two finite sets A and B are mutually exclusive, then

N(A B) = N(A)+ N(B)

Theorem (Probability of the Union of Mutually Exclusive Events):

Addition Counting Principle (Mutually Exclusive Form):

If two finite sets A and B are mutually exclusive, then

N(A B) = N(A)+ N(B)

Theorem (Probability of the Union of Mutually Exclusive Events):

If A and B are mutually exclusive events in the same finite sample space, then

P(A B) = P(A)+ P(B)

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

= 1

36+ 2

36

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

= 1

36+ 2

36 = 3

36

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

= 1

36+ 2

36 = 3

36 = 1

12

Example 1a. If two fair dice are tossed, what is the probability that

the sum is 2 or 3?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum of 2 or 3)

= P(sum of2)+ P(sum of 3)

= 1

36+ 2

36 = 3

36 = 1

12

There is an 8 1/3 % chance of rolling a sum of 2 or 3

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum is even or > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

= 18

36

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

= 18

36 + 30

36

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

= 18

36 + 30

36 − 14

36

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

= 18

36 + 30

36 − 14

36 = 34

36

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

= 18

36 + 30

36 − 14

36 = 34

36 = 17

18

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Example 1b. If two dice are tossed, what is the probability that the

sum will be even or greater than 4?

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

= 18

36 + 30

36 − 14

36 = 34

36 = 17

18

There is a 94 4/9 % chance of rolling an even sum or a sum

greater than 4

P(sum is even or > 4)

= P(sum is even) + P(sum > 4)

Addition Counting Principle (General Form):

Addition Counting Principle (General Form):

For any finite sets A and B,

N(A B) = N(A)+ N(B)− N(A B)

Addition Counting Principle (General Form):

Theorem (Probability of a Union of Events General Form):

For any finite sets A and B,

N(A B) = N(A)+ N(B)− N(A B)

Addition Counting Principle (General Form):

Theorem (Probability of a Union of Events General Form):

For any finite sets A and B,

N(A B) = N(A)+ N(B)− N(A B)

If A and B are any events in the same finite sample space, then

P(A or B) = P(A B) = P(A)+ P(B)− P(A B)

Example 2Thirteen of the 50 states include territory that lies west of the continental divide. Forty-two states include territory that lies east of the continental divide. Is this possible?

Explain.

Example 2Thirteen of the 50 states include territory that lies west of the continental divide. Forty-two states include territory that lies east of the continental divide. Is this possible?

Explain.

Of course this is possible! This just means that some states have the continental divide running right though them!

Example 2Thirteen of the 50 states include territory that lies west of the continental divide. Forty-two states include territory that lies east of the continental divide. Is this possible?

Explain.

Of course this is possible! This just means that some states have the continental divide running right though them!

Bonus: Which states would these be? The FIRST person to post the correct answer AFTER 7 PM on the wiki

under section 7-2 will earn some bonus points!

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

HHHHHTHTHTHH

HTTTHTTTHTTT

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

HHHHHTHTHTHH

HTTTHTTTHTTT

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

HHHHHTHTHTHH

HTTTHTTTHTTT

P(Not all 3 same)

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

HHHHHTHTHTHH

HTTTHTTTHTTT

P(Not all 3 same)

= 6

8

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

HHHHHTHTHTHH

HTTTHTTTHTTT

P(Not all 3 same)

= 6

8 = 3

4

Example 3Three fair coins are tossed. What is the probability that not

all of the coins show the same face?

HHHHHTHTHTHH

HTTTHTTTHTTT

P(Not all 3 same)

= 6

8 = 3

4

There is a 75% chance of

getting not all 3 coins the same

Complementary Events

Complementary Events

When you have events whose union takes up the entire sample space, but the events are mutually exclusive

Complementary Events

When you have events whose union takes up the entire sample space, but the events are mutually exclusive

The complement of R is not R

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(not 6)

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(not 6) = 31

36

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(not 6) = 31

36

There is an 86 1/9 % chance that the sum

will not be 6

Example 4Two dice are tossed. Find the probability that their sum is

not 6.

1,3 1,41,1 1,51,2 1,62,3 2,42,1 2,52,2 2,63,3 3,43,1 3,53,2 3,64,3 4,44,1 4,54,2 4,65,3 5,45,1 5,55,2 5,66,3 6,46,1 6,56,2 6,6

P(not 6) = 31

36

There is an 86 1/9 % chance that the sum

will not be 6

1− P(6) =1− 5

36= 31

36

Theorem (Probability of Complements)

Theorem (Probability of Complements)

In any event E, the complement of E is

P(not E) =1− P(E)

Homework

Homework

p. 437 #1-24