notes 7-2
DESCRIPTION
Addition Counting PrinciplesTRANSCRIPT
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